In this section, before we discuss our main results, let us denote, respectively
$$\begin{aligned}& \Delta ({\tau }):= \int _{0}^{\tau } \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}\mu )}{\mu } \,\mathrm{d} \mu \quad \text{and}\quad \delta ({\tau }):= \int _{\tau }^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}\mu )}{\mu } \,\mathrm{d} \mu , \\& \eta ({\tau }):= \int _{0}^{\tau } \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}(m+1)}\mu )}{\mu } \,\mathrm{d} \mu \quad \text{and}\quad \Omega ({\tau }):= \int _{0}^{\tau } \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )}\mu )}{\mu } \,\mathrm{d} \mu . \end{aligned}$$
2.1 Generalized trapezium inequality
We now derive a new generalized fractional trapezium-type integral inequality using the class of harmonic convex functions. For brevity, we denote in the following \({\Psi }(\tau ):={\frac{1}{\tau }}\).
Theorem 2.1
Let \({\Upsilon }:[{b_{1}},{{b_{2}}}]\rightarrow \mathbb{R}\) be an harmonic convex function, then
$$\begin{aligned} {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)& \leq \frac{1}{2\eta (1)} \biggl[{_{ (\frac{1}{{{b_{2}}}} )^{+}}I_{ \Phi }} {\Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) +{_{ ( \frac{1}{{b_{1}}} )^{-}}I_{\Phi }} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) \biggr] \\ &\leq \frac{[{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})]}{2}, \end{aligned}$$
where \(m\in \mathbb{N}\).
Proof
Since ϒ is an harmonic convex function, then
$$\begin{aligned} {\Upsilon } \biggl(\frac{2xy}{x+y} \biggr)\leq \frac{1}{2} \bigl[{\Upsilon }(x)+{ \Upsilon }(y)\bigr]. \end{aligned}$$
This implies
$$\begin{aligned} &2{\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr) \leq {\Upsilon } \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}} \biggr)+{\Upsilon } \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr). \end{aligned}$$
Multiplying both sides by \(\frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{\tau } )}{{\tau }}\) and integrating with respect to τ on \([0,1]\), we have
$$\begin{aligned} &2{\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr) \int _{0}^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{\tau } )}{{\tau }} \,\mathrm{d} {\tau } \\ &\quad \leq \biggl[ \int _{0}^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{\tau } )}{{\tau }}{ \Upsilon } \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \\ &\qquad{} + \int _{0}^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{\tau } )}{{\tau }}{ \Upsilon } \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \biggr]. \end{aligned}$$
This implies
$$\begin{aligned} &{2\eta (1)} {\Upsilon } \biggl( \frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr) \\ &\quad \leq \int _{0}^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{\tau } )}{{\tau }}{ \Upsilon } \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \\ &\qquad{} + \int _{0}^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{\tau } )}{{\tau }}{ \Upsilon } \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \\ &\quad = \int _{\frac{1}{{{b_{2}}}}}^{ \frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}} \frac{\Phi (\frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}-x )}{ (\frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}-x )}{ \Upsilon \circ \Psi }(x) \,\mathrm{d}x+ \int _{ \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}}^{\frac{1}{{b_{1}}}} \frac{\Phi (x-\frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} )}{ (x-\frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} )}{ \Upsilon \circ \Psi }(x) \,\mathrm{d}x \\ &\quad ={{}_{ (\frac{1}{{{b_{2}}}} )^{+}}I_{\Phi }} {\Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr)+{_{ (\frac{1}{{b_{1}}} )^{-}}I_{\Phi }} {\Upsilon \circ \Psi } \biggl(\frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr). \end{aligned}$$
Now, we prove the second inequality, for this we have
$$\begin{aligned} &{\Upsilon } \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}} \biggr)\leq \frac{m+{\tau }}{m+1}{ \Upsilon }({b_{1}})+ \frac{1-{\tau }}{m+1}{\Upsilon }({{b_{2}}}). \end{aligned}$$
(2.1)
$$\begin{aligned} &{\Upsilon } \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr)\leq \frac{m+{\tau }}{m+1}{ \Upsilon }({{b_{2}}})+ \frac{1-{\tau }}{m+1}{\Upsilon }({b_{1}}). \end{aligned}$$
(2.2)
Adding (2.1) and (2.2) and multiplying both sides by \(\frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{\tau } )}{{\tau }}\) and integrating with respect to τ on \([0,1]\), we have
$$\begin{aligned} & \int _{0}^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{\tau } )}{{\tau }}{ \Upsilon } \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }\\ &\qquad {}+ \int _{0}^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{\tau } )}{{\tau }}{ \Upsilon } \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \\ &\quad \leq {\bigl[{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}}) \bigr]} \int _{0}^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{(m+1){b_{1}{b_{2}}}}{\tau } )}{{\tau }} \,\mathrm{d} {\tau }. \end{aligned}$$
Using generalized fractional integrals, we obtain our second inequality. This completes the proof. □
Corollary 2.1
If we choose \(\Phi ({\tau })={\tau }\) and \(m=1\) in Theorem 2.1, we have
$$\begin{aligned} {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)& \leq \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \int _{ \frac{1}{{{b_{2}}}}}^{\frac{1}{{b_{1}}}}{\Upsilon \circ \Psi }(x) \,\mathrm{d}x\leq \frac{[{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})]}{2}. \end{aligned}$$
Corollary 2.2
If we choose \(\Phi ({\tau })=\frac{{\tau }^{{\alpha }}}{\Gamma (\alpha )}\) in Theorem 2.1, we obtain
$$\begin{aligned} {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)\leq{}& \frac{({b_{1}{b_{2}}}(m+1))^{\alpha }\Gamma (\alpha +1)}{2({{b_{2}}}-{b_{1}})^{\alpha }}\\ &{}\times \biggl[{J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }} {\Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) +{J_{ (\frac{1}{{b_{1}}} )^{-}}} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) \biggr] \\ \leq {}&\frac{[{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})]}{2}. \end{aligned}$$
For \(m=1\), we obtain
$$\begin{aligned} {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)\leq {}& \frac{2^{\alpha -1}({b_{1}{b_{2}}})^{\alpha }\Gamma (\alpha +1)}{({{b_{2}}}-{b_{1}})^{\alpha }}\\ &{}\times \biggl[{J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} \biggr) +{J_{ (\frac{1}{{b_{1}}} )^{-}}} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} \biggr) \biggr] \\ \leq {}&\frac{[{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})]}{2}. \end{aligned}$$
Corollary 2.3
If we choose \(\Phi ({\tau })= \frac{{\tau }^{\frac{\alpha }{k}}}{k\Gamma _{k}(\alpha )}\) in Theorem 2.1, we have
$$\begin{aligned} {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr) \leq {}&\frac{((m+1){b_{1}{b_{2}}})^{\frac{\alpha }{k}}\Gamma _{k}(\alpha +k)}{2({{b_{2}}}-{b_{1}})^{\frac{\alpha }{k}}} \\ &{}\times\biggl[{_{k}J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }} { \Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) +{_{k}J_{ (\frac{1}{{b_{1}}} )^{-}}^{\alpha }} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) \biggr] \\ \leq {}&\frac{[{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})]}{2}. \end{aligned}$$
For \(m=1\), we obtain
$$\begin{aligned} {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr) \leq {}&\frac{2^{\frac{\alpha }{k}-1}({b_{1}{b_{2}}})^{\frac{\alpha }{k}}\Gamma _{k}(\alpha +k)}{({{b_{2}}}-{b_{1}})^{\frac{\alpha }{k}}} \\ &{}\times \biggl[{_{k}J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }} { \Upsilon \circ \Psi } \biggl(\frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} \biggr) +{_{k}J_{ (\frac{1}{{b_{1}}} )^{-}}^{\alpha }} { \Upsilon \circ \Psi } \biggl(\frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} \biggr) \biggr] \\ \leq{}& \frac{[{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})]}{2}. \end{aligned}$$
2.2 Auxiliary results
In this subsection, we derive three new fractional integral identities that will be used in the following.
Lemma 2.2
Let \({\Upsilon }:[{b_{1}},{{b_{2}}}]\rightarrow \mathbb{R}\) be a differentiable function on \(({b_{1}},{{b_{2}}})\) with \({b_{1}}<{{b_{2}}}\) and \(m\in \mathbb{N}\), then
$$\begin{aligned} &\frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{m+1}\\ &\qquad {}- \frac{1}{(m+1)\eta (1)} \biggl[{_{ (\frac{1}{{{b_{2}}}} )^{+}}I_{ \Phi }} {\Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) +{_{ ( \frac{1}{{b_{1}}} )^{-}}I_{\Phi }} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) \biggr] \\ &\quad =\frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\eta (1)} \biggl[ \int _{0}^{1} \frac{\eta ({\tau })}{((m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \\ &\qquad {} - \int _{0}^{1} \frac{\eta ({\tau })}{((1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \biggr]. \end{aligned}$$
Proof
Consider the right-hand side
$$\begin{aligned} I:={}&\frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\eta (1)} \biggl[ \int _{0}^{1} \frac{\eta ({\tau })}{((m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \\ &{} - \int _{0}^{1} \frac{\eta ({\tau })}{((1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \biggr] \\ ={}&\frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\eta (1)}[I_{1}-I_{2}], \end{aligned}$$
where
$$\begin{aligned} I_{1}:={}& \int _{0}^{1} \frac{\eta ({\tau })}{((m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \\ ={}& \frac{\eta (1){\Upsilon }({{b_{2}}})}{(m+1){b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}\\ &{}- \frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(m+1)} \int _{ \frac{1}{{{b_{2}}}}}^{\frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}} \frac{\Phi (\frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}-x )}{\frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}-x}{ \Upsilon \circ \Psi }(x) \,\mathrm{d}x \\ ={}& \frac{\eta (1){\Upsilon }({{b_{2}}})}{(m+1){b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}- \frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(m+1)}{_{ ( \frac{1}{{{b_{2}}}} )^{+}}I_{\Phi }} {\Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr). \end{aligned}$$
Similarly,
$$\begin{aligned} I_{2}:={}& \int _{0}^{1} \frac{\eta ({\tau })}{((1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \\ ={}&{-} \frac{\eta (1){\Upsilon }({b_{1}})}{(m+1){b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}\\ &{}+ \frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(m+1)} \int _{ \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}}^{\frac{1}{{b_{1}}}} \frac{\Phi (x-\frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} )}{x-\frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}}}{ \Upsilon \circ \Psi }(x) \,\mathrm{d}x \\ ={}&{-} \frac{\eta (1){\Upsilon }({b_{1}})}{(m+1){b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}+ \frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(m+1)}{_{ ( \frac{1}{{b_{1}}} )^{-}}I_{\Phi }} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr). \end{aligned}$$
Substituting the values of \(I_{1}\) and \(I_{2}\) in I, we obtain our required result. □
Remark 2.1
If we choose \(m=1\) and \(\Phi ({\tau })={\tau }\), we have
$$\begin{aligned} &\frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{2}- \frac{{b_{1}{b_{2}}}}{({{b_{2}}}-{b_{1}})} \int _{\frac{1}{{{b_{2}}}}}^{ \frac{1}{{b_{1}}}}{\Upsilon \circ \Psi }(x) \,\mathrm{d}x \\ &\quad ={{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} \biggl[ \int _{0}^{1} \frac{{\tau }}{((1+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{\Upsilon }' \biggl(\frac{2{b_{1}{b_{2}}}}{(1+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \\ & \qquad {}- \int _{0}^{1} \frac{{\tau }}{((1-{\tau }){b_{1}}+(1+{\tau }){{b_{2}}})^{2}}{\Upsilon }' \biggl(\frac{2{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(1+{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \biggr]. \end{aligned}$$
Corollary 2.4
If we take \(m=1\) and \(\Phi (\tau )=\frac{{\tau }^{\alpha }}{\Gamma (\alpha )}\) in Lemma 2.2, we obtain
$$\begin{aligned} &\frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{2}\\ &\qquad {}- \frac{2^{\alpha -1}({b_{1}{b_{2}}})^{\alpha }\Gamma (\alpha +1)}{({{b_{2}}}-{b_{1}})^{\alpha }} \biggl[{J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }} {\Upsilon \circ \Psi } \biggl(\frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} \biggr) +J_{ (\frac{1}{{b_{1}}} )^{-}}^{\alpha }{ \Upsilon \circ \Psi } \biggl(\frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} \biggr) \biggr] \\ &\quad ={{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} \biggl[ \int _{0}^{1} \frac{{\tau }^{\alpha }}{((1+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{2{b_{1}{b_{2}}}}{(1+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \\ &\qquad {} - \int _{0}^{1} \frac{{\tau }^{\alpha }}{((1-{\tau }){b_{1}}+(1+{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{2{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \biggr]. \end{aligned}$$
Corollary 2.5
If we choose \(m=1\) and \(\Phi (\tau )=\frac{{\tau }^{\frac{\alpha }{k}}}{k\Gamma _{k}(\alpha )}\) in Lemma 2.2, we obtain
$$\begin{aligned} &\frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{2}\\ &\qquad {}- \frac{2^{\frac{\alpha }{k}-1}({b_{1}{b_{2}}})^{\frac{\alpha }{k}}\Gamma _{k}(\alpha +k)}{({{b_{2}}}-{b_{1}})^{\frac{\alpha }{k}}} \biggl[{_{k}J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }} { \Upsilon \circ \Psi } \biggl(\frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} \biggr) +{_{k}J_{ (\frac{1}{{b_{1}}} )^{-}}^{\alpha }} { \Upsilon \circ \Psi } \biggl(\frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} \biggr) \biggr] \\ &\quad ={{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} \biggl[ \int _{0}^{1} \frac{{\tau }^{\frac{\alpha }{k} }}{((1+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{2{b_{1}{b_{2}}}}{(1+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \\ &\qquad {} - \int _{0}^{1} \frac{{\tau }^{\frac{\alpha }{k}}}{((1-{\tau }){b_{1}}+(1+{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{2{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau } \biggr]. \end{aligned}$$
Lemma 2.3
Let \(\Upsilon :[{b_{1}},{{b_{2}}}]\rightarrow \mathbb{R}\) be a differentiable function on \(({b_{1}},{{b_{2}}})\) with \({b_{1}}<{{b_{2}}}\) and \(\lambda ,\mu \in [0,\infty )\) with \(\lambda +\mu \neq0\), then
$$\begin{aligned} &\frac{\Omega (\lambda )\Upsilon ({{b_{2}}})+\Omega (\mu )\Upsilon ({b_{1}})}{\lambda +\mu }\\ &\qquad {}- \frac{1}{\lambda +\mu } \biggl[{_{ (\frac{1}{{{b_{2}}}} )^{+}}I_{ \Phi }{ \Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} +{_{ (\frac{1}{{b_{1}}} )^{-}}I_{\Phi }{ \Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \biggr] \\ &\quad ={b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \int _{0}^{\lambda } \frac{\Omega ({\tau })}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \Upsilon ' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}}} \biggr) \,\mathrm{d} {\tau } \\ &\qquad {}- \int _{0}^{\mu } \frac{\Omega ({\tau })}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \Upsilon ' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}}} \biggr) \,\mathrm{d} {\tau } \biggr]. \end{aligned}$$
Proof
Consider the right-hand side
$$\begin{aligned} I:={}&{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \int _{0}^{\lambda } \frac{\Omega ({\tau })}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \Upsilon ' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}}} \biggr) \,\mathrm{d} {\tau } \\ &{}- \int _{0}^{\mu } \frac{\Omega ({\tau })}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \Upsilon ' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}}} \biggr) \,\mathrm{d} {\tau } \biggr] \\ ={}&{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})[I_{3}-I_{4}], \end{aligned}$$
(2.3)
where
$$\begin{aligned} I_{3}:={}& \int _{0}^{\lambda } \frac{\Omega ({\tau })}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \Upsilon ' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}}} \biggr) \,\mathrm{d} {\tau } \\ ={}& \frac{\Omega (\lambda )\Upsilon ({{b_{2}}})}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(\lambda +\mu )}\\ &{}- \frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(\lambda +\mu )} \int _{ \frac{1}{{{b_{2}}}}}^{ \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )}} \frac{\Phi (\frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )}-x )}{ (\frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )}-x )}{ \Upsilon \circ \Psi }(x) \,\mathrm{d}x \\ ={}& \frac{\Omega (\lambda )\Upsilon ({{b_{2}}})}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(\lambda +\mu )}- \frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(\lambda +\mu )}{_{ ( \frac{1}{{{b_{2}}}} )^{+}}I_{\Phi }{ \Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)}. \end{aligned}$$
Similarly,
$$\begin{aligned} I_{4}:={}& \int _{0}^{\mu } \frac{\Omega ({\tau })}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \Upsilon ' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}}} \biggr) \,\mathrm{d} {\tau } \\ ={}&{-} \frac{\Omega (\mu )\Upsilon ({b_{1}})}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(\lambda +\mu )}\\ &{}+ \frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(\lambda +\mu )} \int _{ \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )}}^{ \frac{1}{{b_{1}}}} \frac{\Phi (x-\frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} )}{ (x-\frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} )}{ \Upsilon \circ \Psi }(x) \,\mathrm{d}x \\ ={}&{-} \frac{\Omega (\mu )\Upsilon ({{b_{2}}})}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(\lambda +\mu )}- \frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})(\lambda +\mu )}{_{ ( \frac{1}{{b_{1}}} )^{-}}I_{\Phi }{ \Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)}. \end{aligned}$$
Substituting the values of \(I_{3}\) and \(I_{4}\) in (2.3), we obtain our required result. □
Corollary 2.6
Choosing \(\Phi ({\tau })={\tau }\) in Lemma 2.3, we have
$$\begin{aligned} &\frac{\lambda \Upsilon ({{b_{2}}})+\mu \Upsilon ({b_{1}})}{\lambda +\mu }- \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \int _{\frac{1}{{{b_{2}}}}}^{ \frac{1}{{b_{1}}}}{\Upsilon \circ \Psi }(x) \,\mathrm{d}x \\ &\quad ={b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \int _{0}^{\lambda } \frac{{\tau }}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \Upsilon ' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}}} \biggr) \,\mathrm{d} {\tau } \\ &\qquad {}- \int _{0}^{\mu } \frac{{\tau }}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \Upsilon ' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}}} \biggr) \,\mathrm{d} {\tau } \biggr]. \end{aligned}$$
Corollary 2.7
Taking \(\Phi ({\tau })=\frac{{\tau }^{\alpha }}{\Gamma (\alpha )}\) in Lemma 2.3, we obtain
$$\begin{aligned} &\frac{\lambda ^{\alpha }\Upsilon ({{b_{2}}})+\mu ^{\alpha }\Upsilon ({b_{1}})}{\lambda +\mu }- \frac{({b_{1}{b_{2}}})^{\alpha }(\lambda +\mu )^{\alpha -1}\Gamma (\alpha +1)}{({{b_{2}}}-{b_{1}})^{\alpha }} \biggl[{J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }{ \Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \\ &\qquad {}+{J_{ (\frac{1}{{b_{1}}} )^{-}}^{\alpha }{\Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \biggr] \\ &\quad ={b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \int _{0}^{\lambda } \frac{{\tau }^{\alpha }}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \Upsilon ' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}}} \biggr) \,\mathrm{d} {\tau } \\ &\qquad {}- \int _{0}^{\mu } \frac{{\tau }^{\alpha }}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \Upsilon ' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}}} \biggr) \,\mathrm{d} {\tau } \biggr]. \end{aligned}$$
Corollary 2.8
Choosing \(\Phi ({\tau })= \frac{{\tau }^{\frac{\alpha }{k}}}{k\Gamma _{k}(\alpha )}\) in Lemma 2.3, we obtain
$$\begin{aligned} &\frac{\lambda ^{\frac{\alpha }{k}}\Upsilon ({{b_{2}}})+\mu ^{\frac{\alpha }{k}}\Upsilon ({b_{1}})}{\lambda +\mu }- \frac{k({b_{1}{b_{2}}})^{\frac{\alpha }{k}}(\lambda +\mu )^{\frac{\alpha }{k}-1}\Gamma _{k}(\alpha +k)}{({{b_{2}}}-{b_{1}})^{\frac{\alpha }{k}}} \biggl[{_{k}{J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }} { \Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \\ &\qquad {}+{{_{k}J_{ (\frac{1}{{b_{1}}} )^{-}}^{\alpha }} { \Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \biggr] \\ &\quad ={b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \int _{0}^{\lambda } \frac{{\tau }^{\frac{\alpha }{k}}}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \Upsilon ' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}}} \biggr) \,\mathrm{d} {\tau } \\ &\qquad {}- \int _{0}^{\mu } \frac{{\tau }^{\frac{\alpha }{k}}}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \Upsilon ' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}}} \biggr) \,\mathrm{d} {\tau } \biggr]. \end{aligned}$$
Lemma 2.4
Let \({\Upsilon }:[{b_{1}},{{b_{2}}}]\subset (0,+\infty )\rightarrow \mathbb{R}\) be a differentiable mapping on \(({b_{1}},{{b_{2}}})\) with \({b_{1}}<{{b_{2}}}\), then
$$\begin{aligned} &{\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{1}{2\Delta (1)} \biggl[_{\frac{1}{{{b_{2}}}}^{+}}I_{\Phi }{{ \Upsilon }\circ {\Psi }} \biggl( \frac{1}{{b_{1}}} \biggr)+ _{ \frac{1}{{b_{1}}}^{-}}I_{\Phi }{{\Upsilon } \circ {\Psi }} \biggl( \frac{1}{{{b_{2}}}} \biggr) \biggr]\\ &\quad = \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2\Delta (1)}\sum_{j=1}^{4}M_{j}, \end{aligned}$$
where
$$\begin{aligned}& M_{1}:= \int _{0}^{\frac{1}{2}} \frac{\Delta ({\tau })}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& M_{2}:= \int _{0}^{\frac{1}{2}} \frac{ (-\Delta ({\tau }) )}{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+{\tau {b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& M_{3}:= \int _{\frac{1}{2}}^{1} \frac{ (-\delta ({\tau }) )}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& M_{4}:= \int _{\frac{1}{2}}^{1} \frac{\delta ({\tau })}{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+{\tau {b_{2}}}} \biggr) \,\mathrm{d} {\tau }. \end{aligned}$$
Proof
Integrating by parts \(M_{i}\) for \(i=1,2,3,4\), and changing the variables, we have
$$\begin{aligned}& \begin{aligned} M_{1}={}&\frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\Upsilon } \biggl( \frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr) \int _{0}^{ \frac{1}{2}} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}\mu )}{\mu } \,\mathrm{d} \mu \\ &{}-\frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} \int _{0}^{ \frac{1}{2}} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}{\tau } )}{{\tau }}{ \Upsilon } \biggl( \frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \end{aligned} \\& \begin{aligned} M_{2}={}&\frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\Upsilon } \biggl( \frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr) \int _{0}^{ \frac{1}{2}} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}\mu )}{\mu } \,\mathrm{d} \mu \\ &{}-\frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} \int _{0}^{ \frac{1}{2}} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}{\tau } )}{{\tau }}{ \Upsilon } \biggl( \frac{{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+{\tau {b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \end{aligned} \\& \begin{aligned} M_{3}={}&\frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\Upsilon } \biggl( \frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr) \int _{\frac{1}{2}}^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}\mu )}{\mu } \,\mathrm{d} \mu \\ &{}-\frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} \int _{ \frac{1}{2}}^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}{\tau } )}{{\tau }}{ \Upsilon } \biggl( \frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \end{aligned} \\& \begin{aligned} M_{4}={}&\frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\Upsilon } \biggl( \frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr) \int _{\frac{1}{2}}^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}\mu )}{\mu } \,\mathrm{d} \mu \\ &{}-\frac{1}{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} \int _{ \frac{1}{2}}^{1} \frac{\Phi (\frac{{{b_{2}}}-{b_{1}}}{{b_{1}{b_{2}}}}{\tau } )}{{\tau }}{ \Upsilon } \biggl( \frac{{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+{\tau {b_{2}}}} \biggr) \,\mathrm{d} {\tau }. \end{aligned} \end{aligned}$$
Adding \(M_{1}\), \(M_{2}\), \(M_{3}\) and \(M_{4}\) and multiplying by the factor \(\frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2\Delta (1)}\), we obtain our required result. □
Corollary 2.9
Taking \(\Phi ({\tau })={\tau }\) in Lemma 2.4, then
$$\begin{aligned} &{\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \int _{b_{1}}^{{b_{2}}} \frac{{\Upsilon }(x)}{x^{2}} \,\mathrm{d}x= \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2}\sum_{j=1}^{4}L_{j}, \end{aligned}$$
where
$$\begin{aligned}& L_{1}:= \int _{0}^{\frac{1}{2}} \frac{{\tau }}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{\Upsilon }' \biggl(\frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& L_{2}:= \int _{0}^{\frac{1}{2}} \frac{{-\tau }}{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}}{\Upsilon }' \biggl(\frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& L_{3}:= \int _{\frac{1}{2}}^{1} \frac{-{\tau }}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{\Upsilon }' \biggl(\frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& L_{4}:= \int _{\frac{1}{2}}^{1} \frac{{\tau }}{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}}{\Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+{\tau {b_{2}}}} \biggr) \,\mathrm{d} {\tau }. \end{aligned}$$
Corollary 2.10
Choosing \(\Phi ({\tau })=\frac{{\tau }^{\alpha }}{\Gamma (\alpha )}\) in Lemma 2.4, then
$$\begin{aligned} &{\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{\Gamma (\alpha +1)}{2} \biggl( \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \biggr)^{\alpha } \biggl[J_{ \frac{1}{{{b_{2}}}}^{+}}^{\alpha }{{ \Upsilon }\circ {\Psi }} \biggl( \frac{1}{{b_{1}}} \biggr) +J_{\frac{1}{{b_{1}}}^{-}}^{\alpha }{{ \Upsilon }\circ {\Psi }} \biggl( \frac{1}{{{b_{2}}}} \biggr) \biggr] \\ &\quad =\frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2}\sum_{j=5}^{8}L_{j}, \end{aligned}$$
where
$$\begin{aligned}& L_{5}:= \int _{0}^{\frac{1}{2}} \frac{{\tau }^{\alpha }}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{\Upsilon }' \biggl(\frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& L_{6}:= \int _{0}^{\frac{1}{2}} \frac{(-{\tau })^{\alpha }}{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+{\tau {b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& L_{7}:= \int _{\frac{1}{2}}^{1} \frac{(-{\tau })^{\alpha }}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& L_{8}:= \int _{\frac{1}{2}}^{1} \frac{{\tau }^{\alpha }}{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}}{\Upsilon }' \biggl(\frac{{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+{\tau {b_{2}}}} \biggr) \,\mathrm{d} {\tau }. \end{aligned}$$
Corollary 2.11
Taking \(\Phi ({\tau })= \frac{{\tau }^{\frac{\alpha }{k}}}{k\Gamma _{k}(\alpha )}\) in Lemma 2.4, then
$$\begin{aligned} &{\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{\Gamma _{k}(\alpha +k)}{2} \biggl( \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \biggr)^{\frac{\alpha }{k}} \biggl[ _{k}J_{\frac{1}{{{b_{2}}}}^{+}}^{\alpha }{{ \Upsilon }\circ { \Psi }} \biggl(\frac{1}{{b_{1}}} \biggr) + _{k}J_{\frac{1}{{b_{1}}}^{-}}^{ \alpha }{{\Upsilon }\circ {\Psi }} \biggl(\frac{1}{{{b_{2}}}} \biggr) \biggr] \\ &\quad =\frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2}\sum_{j=9}^{12}L_{j}, \end{aligned}$$
where
$$\begin{aligned}& L_{9}:= \int _{0}^{\frac{1}{2}} \frac{{\tau }^{\frac{\alpha }{k}}}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& L_{10}:= \int _{0}^{\frac{1}{2}} \frac{(-{\tau })^{\frac{\alpha }{k}}}{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+{\tau {b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& L_{11}:= \int _{\frac{1}{2}}^{1} \frac{(-{\tau })^{\frac{\alpha }{k}}}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& L_{12}:= \int _{\frac{1}{2}}^{1} \frac{{\tau }^{\frac{\alpha }{k}}}{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+{\tau {b_{2}}}} \biggr) \,\mathrm{d} {\tau }. \end{aligned}$$
Corollary 2.12
Choosing \(\Phi ({\tau })=\frac{{\tau }}{\alpha }\exp (-A{\tau } )\) in Lemma 2.4with \(A=\frac{1-\alpha }{\alpha }\) and \(\alpha \in (0,1]\), then
$$\begin{aligned} &{\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{1-\alpha }{2(1-\exp (-A))} \biggl( \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \biggr)^{\alpha } \biggl[I_{ \frac{1}{{{b_{2}}}}^{+}}^{\alpha }{{ \Upsilon }\circ {\Psi }} \biggl( \frac{1}{{b_{1}}} \biggr) +I_{\frac{1}{{b_{1}}}^{-}}^{\alpha }{{ \Upsilon }\circ {\Psi }} \biggl( \frac{1}{{{b_{2}}}} \biggr) \biggr] \\ &\quad =\frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2(1-\exp (-A))}\sum_{j=13}^{16}L_{j}, \end{aligned}$$
where
$$\begin{aligned}& L_{13}:= \int _{0}^{\frac{1}{2}} \frac{[\exp (-A{\tau })-1]}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& L_{14}:= \int _{0}^{\frac{1}{2}} \frac{[1-\exp (-A{\tau })]}{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+{\tau {b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& L_{15}:= \int _{\frac{1}{2}}^{1} \frac{[\exp (-A(1-{\tau }))-\exp (-A{\tau })]}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{{\tau b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \,\mathrm{d} {\tau }, \\& L_{16}:= \int _{\frac{1}{2}}^{1} \frac{[\exp (-A{\tau })-\exp (-A(1-{\tau }))]}{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}}{ \Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+{\tau {b_{2}}}} \biggr) \,\mathrm{d} {\tau }. \end{aligned}$$
2.3 Further results
Now, utilizing auxiliary results obtained in the previous subsection, we derive some further generalized fractional trapezium-like inequalities using the class of harmonic convex functions.
Theorem 2.5
Let \({\Upsilon }:[{b_{1}},{{b_{2}}}]\rightarrow \mathbb{R}\) be a continuous function on \(({b_{1}},{{b_{2}}})\) with \({b_{1}}<{{b_{2}}}\) and \(|{\Upsilon }'|^{q}\) be an harmonic convex function with \(\frac{1}{p}+\frac{1}{q}=1\), then
$$\begin{aligned} & \biggl\vert \frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{m+1}\\ &\qquad {}- \frac{1}{(m+1)\eta (1)} \biggl[{_{ (\frac{1}{{{b_{2}}}} )^{+}}I_{ \Phi }} {\Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) +{_{ ( \frac{1}{{b_{1}}} )^{-}}I_{\Phi }} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\eta (1)} \biggl[\pi _{1}^{ \frac{1}{p}} \biggl( \int _{0}^{1}\eta ^{q}({\tau }) \biggl( \frac{1-{\tau }}{m+1} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+\frac{m+{\tau }}{m+1} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} {\tau } \biggr)^{ \frac{1}{q}} \\ &\qquad {}+ \pi _{2}^{\frac{1}{p}} \biggl( \int _{0}^{1}\eta ^{q}({\tau }) \biggl(\frac{m+{\tau }}{m+1} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+ \frac{1-{\tau }}{m+1} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} { \tau } \biggr)^{\frac{1}{q}} \biggr], \end{aligned}$$
where
$$\begin{aligned} &\pi _{1} := \frac{(m{b_{1}}+{{b_{2}}})^{1-2p}}{({{b_{2}}}-{b_{1}})(1-2p)} \biggl[1- \biggl( \frac{(m+1){b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)^{1-2p} \biggr], \\ &\pi _{2} := \frac{({b_{1}}+m{{b_{2}}})^{1-2p}}{({{b_{2}}}-{b_{1}})(1-2p)} \biggl[ \biggl( \frac{(m+1){{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)^{1-2p}-1 \biggr]. \end{aligned}$$
Proof
Using Lemma 2.2, the modulus property, Hölder’s inequality and the harmonic convexity of \(|{\Upsilon }'|^{q}\), we have
$$\begin{aligned} & \biggl\vert \frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{m+1}\\ &\qquad {}- \frac{1}{(m+1)\eta (1)} \biggl[{_{ (\frac{1}{{{b_{2}}}} )^{+}}I_{ \Phi }} {\Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) +{_{ ( \frac{1}{{b_{1}}} )^{-}}I_{\Phi }} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\eta (1)} \biggl[ \int _{0}^{1} \frac{\eta ({\tau })}{((m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}})^{2}} \biggl\vert {\Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \biggr\vert \,\mathrm{d} {\tau } \\ &\qquad {} + \int _{0}^{1} \frac{\eta ({\tau })}{((1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}})^{2}} \biggl\vert {\Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}} \biggr) \biggr\vert \,\mathrm{d} {\tau } \biggr] \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\eta (1)} \biggl[ \biggl( \int _{0}^{1}\bigl((m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}\bigr)^{-2p} \,\mathrm{d} { \tau } \biggr)^{\frac{1}{p}}\\ &\qquad {}\times \biggl( \int _{0}^{1}{\eta ^{p}({\tau })} \biggl\vert {\Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \biggr\vert ^{q} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{q}} \\ & \qquad {}+ \biggl( \int _{0}^{1}\bigl((1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}\bigr)^{-2p} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{p}}\\ &\qquad {}\times \biggl( \int _{0}^{1}{\eta ^{q}({ \tau })} \biggl\vert {\Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}} \biggr) \biggr\vert ^{q} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{q}} \biggr] \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\eta (1)} \biggl[ \biggl( \int _{0}^{1}\bigl((m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}\bigr)^{-2p} \,\mathrm{d} { \tau } \biggr)^{\frac{1}{p}} \\ &\qquad {}\times\biggl( \int _{0}^{1}\eta ^{q}({\tau }) \biggl(\frac{1-{\tau }}{m+1} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+ \frac{m+{\tau }}{m+1} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} { \tau } \biggr)^{\frac{1}{q}} \\ &\qquad {} + \biggl( \int _{0}^{1}\bigl((1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}\bigr)^{-2p} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{p}}\\ &\qquad {}\times \biggl( \int _{0}^{1}\eta ^{q}({ \tau }) \biggl(\frac{m+{\tau }}{m+1} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+ \frac{1-{\tau }}{m+1} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} { \tau } \biggr)^{\frac{1}{q}} \biggr]. \end{aligned}$$
After simple calculations, we obtain our required result. □
Corollary 2.13
Choosing \(\Phi ({\tau })={\tau }\) in Theorem 2.5, we have
$$\begin{aligned} & \biggl\vert \frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{m+1}- \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \int _{\frac{1}{{{b_{2}}}}}^{ \frac{1}{{b_{1}}}}{\Upsilon \circ \Psi }(x) \,\mathrm{d}x \biggr\vert \\ &\quad \leq {{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}\\ &\qquad {}\times \biggl[\pi _{1}^{\frac{1}{p}} \biggl(\frac{1}{(m+1)(q+1)(q+2)} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+ \frac{m(q+2)+(q+1)}{(m+1)(q+1)(q+2)} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad {}+ \pi _{2}^{\frac{1}{p}} \biggl(\frac{1}{(m+1)(q+1)(q+2)} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\frac{m(q+2)+(q+1)}{(m+1)(q+1)(q+2)} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} \biggr], \end{aligned}$$
Corollary 2.14
Taking \(\Phi ({\tau })=\frac{{\tau }^{\alpha }}{\Gamma (\alpha )}\) in Theorem 2.5, we obtain
$$\begin{aligned} &\biggl\vert \frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{m+1}- \frac{(m+1)^{\alpha -1}({b_{1}{b_{2}}})^{\alpha }\Gamma (\alpha +1)}{({{b_{2}}}-{b_{1}})^{\alpha }} \biggl[{J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }} {\Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{{b_{1}{b_{2}}}(m+1)} \biggr) \\ &\qquad {}+J_{ (\frac{1}{{b_{1}}} )^{-}}^{\alpha }{ \Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) \biggr]\biggr\vert \\ &\quad \leq {{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} \biggl[\pi _{1}^{\frac{1}{p}} \biggl(\frac{1}{(m+1)({\alpha }q+1)({\alpha }q+2)} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}\\ &\qquad {}+ \frac{m({\alpha }q+2)+({\alpha }q+1)}{(m+1)({\alpha }q+1)({\alpha }q+2)} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr)^{ \frac{1}{q}} \\ &\qquad {}+ \pi _{2}^{\frac{1}{p}} \biggl( \frac{1}{(m+1)({\alpha } q+1)({\alpha }q+2)} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}\\ &\qquad {}+ \frac{m({\alpha }q+2)+({\alpha }q+1)}{(m+1)({\alpha }q+1)({\alpha }q+2)} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} \biggr], \end{aligned}$$
where \(\pi _{1}\) and \(\pi _{2}\) are already defined.
Corollary 2.15
Choosing \(\Phi ({\tau })= \frac{{\tau }^{\frac{\alpha }{k}}}{k\Gamma _{k}(\alpha )}\) in Theorem 2.5, we obtain
$$\begin{aligned} &\biggl\vert \frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{m+1}- \frac{(m+1)^{\frac{\alpha }{k}-1}({b_{1}{b_{2}}})^{\frac{\alpha }{k}}\Gamma _{k}(\alpha +k)}{({{b_{2}}}-{b_{1}})^{\frac{\alpha }{k}}} \biggl[{_{k}J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }} { \Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{{b_{1}{b_{2}}}(m+1)} \biggr) \\ &\qquad {} +{_{k}J_{ (\frac{1}{{b_{1}}} )^{-}}^{\alpha }} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) \biggr]\biggr\vert \\ &\quad \leq {{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} \biggl[\pi _{1}^{\frac{1}{p}} \biggl(\frac{k}{(m+1)({\alpha }q+k)({\alpha }q+2k)} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}\\ &\qquad {}+ \frac{km({\alpha }q+2k)+k({\alpha }q+k)}{(m+1)({\alpha }q+k)({\alpha }q+2k)} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} \\ &\qquad {}+ \pi _{2}^{\frac{1}{p}} \biggl( \frac{k}{(m+1)({\alpha } q+k)({\alpha }q+2k)} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}\\ &\qquad {}+ \frac{mk({\alpha }q+2k)+(k{\alpha }q+k)}{(m+1)({\alpha }q+k)({\alpha }q+2k)} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} \biggr]. \end{aligned}$$
Theorem 2.6
Let \({\Upsilon }:[{b_{1}},{{b_{2}}}]\rightarrow \mathbb{R}\) be a continuous function on \(({b_{1}},{{b_{2}}})\) with \({b_{1}}<{{b_{2}}}\) and \(|{\Upsilon }'|^{q}\) be an harmonic convex function with \(q\geq 1\), then
$$\begin{aligned} & \biggl\vert \frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{m+1}\\ &\qquad {}- \frac{1}{(m+1)\eta (1)} \biggl[{_{ (\frac{1}{{{b_{2}}}} )^{+}}I_{ \Phi }} {\Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) +{_{ ( \frac{1}{{b_{1}}} )^{-}}I_{\Phi }} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\eta (1)}\biggl[ \biggl( \int _{0}^{1}\eta ({\tau }) \bigl((m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}\bigr)^{-2} \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}} \\ &\qquad {}\times \biggl( \int _{0}^{1}{\eta ({\tau })}\bigl((m+{\tau }){b_{1}}+(1-{ \tau }){{b_{2}}}\bigr)^{-2} \biggl(\frac{1-{\tau }}{m+1} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+ \frac{m+{\tau }}{m+1} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} { \tau } \biggr)^{\frac{1}{q}} \\ &\qquad {}+ \biggl( \int _{0}^{1}\eta ({\tau }) \bigl((1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}\bigr)^{-2} \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}} \\ &\qquad {}\times \biggl( \int _{0}^{1}{\eta ({\tau })}\bigl((1-{\tau }){b_{1}}+(m+{ \tau }){{b_{2}}}\bigr)^{-2}\\ &\qquad {}\times \biggl(\frac{m+{\tau }}{m+1} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+ \frac{1-{\tau }}{m+1} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} { \tau } \biggr)^{\frac{1}{q}}\biggr]. \end{aligned}$$
Proof
Using Lemma 2.2, the modulus property, the power mean inequality and the convexity of \(|{\Upsilon }'|^{q}\), we have
$$\begin{aligned} & \biggl\vert \frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{m+1}\\ &\qquad {}- \frac{1}{(m+1)\eta (1)} \biggl[{_{ (\frac{1}{{{b_{2}}}} )^{+}}I_{ \Phi }} {\Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) +{_{ ( \frac{1}{{b_{1}}} )^{-}}I_{\Phi }} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\eta (1)}\biggl[ \int _{0}^{1} \frac{\eta ({\tau })}{((m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}})^{2}} \biggl\vert {\Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \biggr\vert \,\mathrm{d} {\tau } \\ &\qquad {}+ \int _{0}^{1} \frac{\eta ({\tau })}{((1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}})^{2}} \biggl\vert {\Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}} \biggr) \biggr\vert \,\mathrm{d} {\tau }\biggr] \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\eta (1)}\biggl[ \biggl( \int _{0}^{1}\eta ({\tau }) \bigl((m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}\bigr)^{-2} \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}} \\ &\qquad {}\times \biggl( \int _{0}^{1}{\eta ({\tau })}\bigl((m+{ \tau }){b_{1}}+(1-{\tau }){{b_{2}}}\bigr)^{-2} \biggl\vert {\Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}} \biggr) \biggr\vert ^{q} \,\mathrm{d} { \tau } \biggr)^{\frac{1}{q}} \\ &\qquad {}+ \biggl( \int _{0}^{1}\eta ({\tau }) \bigl((1-{\tau }){b_{1}}+(m+{ \tau }){{b_{2}}}\bigr)^{-2} \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}} \\ &\qquad {} \times \biggl( \int _{0}^{1}{\eta ({\tau })}\bigl((1-{ \tau }){b_{1}}+(m+{\tau }){{b_{2}}}\bigr)^{-2} \biggl\vert {\Upsilon }' \biggl( \frac{(m+1){b_{1}{b_{2}}}}{(1-{\tau }){b_{1}}+(m+{\tau }){{b_{2}}}} \biggr) \biggr\vert ^{q} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{q}}\biggr] \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{\eta (1)}\biggl[ \biggl( \int _{0}^{1}\eta ({\tau }) \bigl((m+{\tau }){b_{1}}+(1-{\tau }){{b_{2}}}\bigr)^{-2} \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}} \\ &\qquad {} \times \biggl( \int _{0}^{1}{\eta ({\tau })}\bigl((m+{ \tau }){b_{1}}+(1-{\tau }){{b_{2}}}\bigr)^{-2} \biggl(\frac{1-{\tau }}{m+1} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q}+\frac{m+{\tau }}{m+1} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} {\tau } \biggr)^{\frac{1}{q}} \\ &\qquad {} + \biggl( \int _{0}^{1}\eta ({\tau }) \bigl((1-{\tau }){b_{1}}+(m+{ \tau }){{b_{2}}}\bigr)^{-2} \biggr)^{1-\frac{1}{q}} \\ &\qquad {}\times \biggl( \int _{0}^{1}{\eta ({\tau })}\bigl((1-{ \tau }){b_{1}}+(m+{\tau }){{b_{2}}}\bigr)^{-2}\\ &\qquad {}\times \biggl(\frac{m+{\tau }}{m+1} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q}+\frac{1-{\tau }}{m+1} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} {\tau } \biggr)^{\frac{1}{q}}\biggr]. \end{aligned}$$
After simple calculations, we obtain our required result. □
Corollary 2.16
If we take \(\Phi ({\tau })={\tau }\) in Theorem 2.6, we have
$$\begin{aligned} & \biggl\vert \frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{m+1}- \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \int _{\frac{1}{{{b_{2}}}}}^{ \frac{1}{{b_{1}}}}{\Upsilon \circ \Psi }(x) \,\mathrm{d}x \biggr\vert \\ &\quad \leq {{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} \bigl[\pi _{3}^{1- \frac{1}{q}} \bigl(\pi _{4} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+\pi _{5} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}\\ &\qquad{}+\pi _{6}^{1- \frac{1}{q}} \bigl(\pi _{7} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+\pi _{8} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr], \end{aligned}$$
where
$$\begin{aligned} &\pi _{3} :=\frac{(m{b_{1}}+{{b_{2}}})^{-2}}{2}{_{2}F_{1} \biggl(2,2,3, \frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\pi _{4} :=\frac{(m{b_{1}}+{{b_{2}}})^{-2}}{6(m+1)}{_{2}F_{1} \biggl(2,2,4, \frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\pi _{5} :=\frac{m(m{b_{1}}+{{b_{2}}})^{-2}}{2(m+1)}{_{2}F_{1} \biggl(2,2,3, \frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)}+ \frac{(m{b_{1}}+{{b_{2}}})^{-2}}{6(m+1)}{_{2}F_{1} \biggl(2,3,4, \frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\pi _{6} :=\frac{({b_{1}}+m{{b_{2}}})^{-2}}{2}{_{2}F_{1} \biggl(2,2,3, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)}, \\ &\pi _{7} :=\frac{m({b_{1}}+m{{b_{2}}})^{-2}}{2(m+1)}{_{2}F_{1} \biggl(2,2,3, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)}+ \frac{({b_{1}}+m{{b_{2}}})^{-2}}{6(m+1)}{_{2}F_{1} \biggl(2,3,4, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)}, \\ &\pi _{8} :=\frac{({b_{1}}+m{{b_{2}}})^{-2}}{6(m+1)}{_{2}F_{1} \biggl(2,2,4, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)}. \end{aligned}$$
Corollary 2.17
If we choose \(\Phi ({\tau })=\frac{{\tau }^{\alpha }}{\Gamma (\alpha )}\) in Theorem 2.6, we obtain
$$\begin{aligned} &\biggl\vert \frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{m+1}- \frac{(m+1)^{\alpha -1}({b_{1}{b_{2}}})^{\alpha }\Gamma (\alpha +1)}{({{b_{2}}}-{b_{1}})^{\alpha }} \biggl[{J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }} {\Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{{b_{1}{b_{2}}}(m+1)} \biggr) \\ &\qquad {}+{J_{ (\frac{1}{{b_{1}}} )^{-}}^{\alpha }} { \Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) \biggr]\biggr\vert \\ &\quad \leq {{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} \bigl[\pi _{9}^{1- \frac{1}{q}} \bigl(\pi _{10} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+\pi _{11} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}\\ &\qquad{}+\pi _{12}^{1- \frac{1}{q}} \bigl(\pi _{13} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+\pi _{14} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr], \end{aligned}$$
where
$$\begin{aligned} &\pi _{9} :=\frac{(m{b_{1}}+{{b_{2}}})^{-2}}{\alpha +1}{_{2}F_{1} \biggl(2,\alpha +1,\alpha +2, \frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\pi _{10} := \frac{(m{b_{1}}+{{b_{2}}})^{-2}}{(\alpha +2)(\alpha +1)(m+1)}{_{2}F_{1} \biggl(2,\alpha +1,\alpha +3, \frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)}, \\ & \begin{aligned} \pi _{11} :={}&\frac{m(m{b_{1}}+{{b_{2}}})^{-2}}{(\alpha +1)(m+1)}{_{2}F_{1} \biggl(2,\alpha +1,\alpha +2, \frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)} \\ &{} + \frac{(m{b_{1}}+{{b_{2}}})^{-2}}{(\alpha +2)(\alpha +1)(m+1)}{_{2}F_{1} \biggl(2,\alpha +2, \alpha +3, \frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)}, \end{aligned} \\ &\pi _{12} :=\frac{({b_{1}}+m{{b_{2}}})^{-2}}{\alpha +1}{_{2}F_{1} \biggl(2,\alpha +1,\alpha +2, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)}, \\ & \begin{aligned} \pi _{13} :={}&\frac{m({b_{1}}+m{{b_{2}}})^{-2}}{(\alpha +1)(m+1)}{_{2}F_{1} \biggl(2,\alpha +1,\alpha +2, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)} \\ & {}+ \frac{({b_{1}}+m{{b_{2}}})^{-2}}{(\alpha +2)(\alpha +1)(m+1)} {_{2}F_{1} \biggl(2,\alpha +2,\alpha +3, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)}, \end{aligned} \\ &\pi _{14} := \frac{({b_{1}}+m{{b_{2}}})^{-2}}{(\alpha +2)(\alpha +1)(m+1)}{_{2}F_{1} \biggl(2,\alpha +1,\alpha +3, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)}. \end{aligned}$$
Corollary 2.18
If we take \(\Phi ({\tau })= \frac{{\tau }^{\frac{\alpha }{k}}}{k\Gamma _{k}(\alpha )}\) in Theorem 2.6, we obtain
$$\begin{aligned} &\biggl\vert \frac{{\Upsilon }({b_{1}})+{\Upsilon }({{b_{2}}})}{m+1}- \frac{(m+1)^{\frac{\alpha }{k}-1}({b_{1}{b_{2}}})^{\frac{\alpha }{k}}\Gamma _{k}(\alpha +k)}{({{b_{2}}}-{b_{1}})^{\frac{\alpha }{k}}} \biggl[{_{k}J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }} { \Upsilon \circ \Psi } \biggl( \frac{m{b_{1}}+{{b_{2}}}}{{b_{1}{b_{2}}}(m+1)} \biggr) \\ & \qquad {}+{_{k}J_{ (\frac{1}{{b_{1}}} )^{-}}^{\alpha }} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+m{{b_{2}}}}{(m+1){b_{1}{b_{2}}}} \biggr) \biggr]\biggr\vert \\ &\quad \leq {{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})} \bigl[\pi _{15}^{1- \frac{1}{q}} \bigl(\pi _{16} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+\pi _{17} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}\\ &\qquad{}+\pi _{18}^{1- \frac{1}{q}} \bigl(\pi _{19} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+\pi _{20} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr], \end{aligned}$$
where
$$\begin{aligned} &\pi _{15} :=\frac{k(m{b_{1}}+{{b_{2}}})^{-2}}{\alpha +k}{_{2}F_{1,k} \biggl(2k,\alpha +k,\alpha +2k, \frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\pi _{16} := \frac{k(m{b_{1}}+{{b_{2}}})^{-2}}{(\alpha +2k)(\alpha +k)(m+1)}{_{2}F_{1,k} \biggl(2k,\alpha +k,\alpha +3k, \frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)}, \\ & \begin{aligned} \pi _{17} :={}&\frac{km(m{b_{1}}+{{b_{2}}})^{-2}}{(\alpha +k)(m+1)}{_{2}F_{1,k} \biggl(2k,\alpha +k,\alpha +2k, \frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)} \\ &{} + \frac{k(m{b_{1}}+{{b_{2}}})^{-2}}{(\alpha +2k)(\alpha +k)(m+1)}{_{2}F_{1,k} \biggl(2k,\alpha +2k,\alpha +3k, \frac{{{b_{2}}}-{b_{1}}}{m{b_{1}}+{{b_{2}}}} \biggr)}, \end{aligned} \\ &\pi _{18} :=\frac{k({b_{1}}+m{{b_{2}}})^{-2}}{\alpha +k}{_{2}F_{1,k} \biggl(2k,\alpha +k,\alpha +2k, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)}, \\ & \begin{aligned} \pi _{19} :={}&\frac{km({b_{1}}+m{{b_{2}}})^{-2}}{(\alpha +k)(m+1)}{_{2}F_{1,k} \biggl(2k,\alpha +k,\alpha +2k, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)} \\ &{} + \frac{k({b_{1}}+m{{b_{2}}})^{-2}}{(\alpha +2k)(\alpha +k)(m+1)} {_{2}F_{1,k} \biggl(2k,\alpha +2k,\alpha +3k, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)}, \end{aligned} \\ &\pi _{20} := \frac{k({b_{1}}+m{{b_{2}}})^{-2}}{(\alpha +2k)(\alpha +k)(m+1)}{_{2}F_{1,k} \biggl(2k,\alpha +k,\alpha +3k, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+m{{b_{2}}}} \biggr)}. \end{aligned}$$
Theorem 2.7
Let \({\Upsilon }:[{b_{1}},{{b_{2}}}]\rightarrow \mathbb{R}\) be a continuous function on \(({b_{1}},{{b_{2}}})\) with \({b_{1}}<{{b_{2}}}\) and \(|{\Upsilon }'|^{q}\) be an harmonic convex function with \(\frac{1}{p}+\frac{1}{q}=1\) and \(\lambda ,\mu \in [0,\infty )\) with \(\lambda +\mu \neq0\), then
$$\begin{aligned} & \biggl\vert \frac{\Omega (\lambda ){\Upsilon }({{b_{2}}})+\Omega (\mu ){\Upsilon }({b_{1}})}{\lambda +\mu }\\ &\qquad{}- \frac{1}{\lambda +\mu } \biggl[{_{ (\frac{1}{{{b_{2}}}} )^{+}}I_{ \Phi }{\Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} +{_{ (\frac{1}{{b_{1}}} )^{-}}I_{\Phi }{\Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \biggr] \biggr\vert \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \biggl( \int _{0}^{ \lambda }{\Omega ^{p}({\tau })} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{p}} \bigl(\sigma _{1} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{2} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}\\ &\qquad {}+ \biggl( \int _{0}^{\mu }{\Omega ^{p}({\tau })} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{p}} \bigl(\sigma _{3} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{4} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \biggr], \end{aligned}$$
where
$$\begin{aligned}& \begin{aligned} \sigma _{1}& := \int _{0}^{\lambda }\frac{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{-2q}(\mu +{\tau })}{\lambda +\mu } \,\mathrm{d} { \tau } \\ &= \frac{\mu (\lambda {{b_{2}}}+\mu {b_{1}})^{1-2q}-(\lambda +\mu )((\lambda +\mu ){b_{1}})^{1-2q}}{(\lambda +\mu )({{b_{2}}}-{b_{1}})(1-2q)}- \frac{((\lambda +\mu ){b_{1}})^{2-2q}-(\lambda {{b_{2}}}+\mu {b_{1}})^{2-2q}}{(\lambda +\mu )({{b_{2}}}-{b_{1}})^{2}(1-2q)(2-2q)}, \end{aligned}\\& \begin{aligned} \sigma _{2}& := \int _{0}^{\lambda }\frac{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{-2q}(\lambda -{\tau })}{\lambda +\mu } \,\mathrm{d} { \tau } \\ &= \frac{\lambda (\lambda {{b_{2}}}+\mu {b_{1}})^{1-2q}}{(\lambda +\mu )({{b_{2}}}-{b_{1}})(1-2q)}+ \frac{((\lambda +\mu ){b_{1}})^{2-2q}-(\lambda {{b_{2}}}+\mu {b_{1}})^{2-2q}}{(\lambda +\mu )({{b_{2}}}-{b_{1}})^{2}(1-2q)(2-2q)}, \end{aligned}\\& \begin{aligned} \sigma _{3}& := \int _{0}^{\mu }\frac{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{-2q}(\mu -{\tau })}{\lambda +\mu } \,\mathrm{d} { \tau } \\ &= \frac{((\lambda +\mu ){{b_{2}}})^{2-2q}-(\lambda {{b_{2}}}+\mu {b_{1}})^{2-2q}}{(\lambda +\mu )({{b_{2}}}-{b_{1}})^{2}(1-2q)(2-2q)}- \frac{\mu (\lambda {{b_{2}}}+\mu {b_{1}})^{1-2q}}{(\lambda +\mu )({{b_{2}}}-{b_{1}})(1-2q)}, \end{aligned}\\& \begin{aligned} \sigma _{4}& := \int _{0}^{\mu }\frac{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{-2q}(\lambda +{\tau })}{\lambda +\mu } \,\mathrm{d} { \tau } \\ &= \frac{(\lambda +\mu )((\lambda +\mu ){{b_{2}}})^{1-2q}-\lambda (\lambda {{b_{2}}}+\mu {b_{1}})^{1-2q}}{(\lambda +\mu )({{b_{2}}}-{b_{1}})(1-2q)}- \frac{((\lambda +\mu ){{b_{2}}})^{2-2q}-(\lambda {{b_{2}}}+\mu {b_{1}})^{2-2q}}{(\lambda +\mu )({{b_{2}}}-{b_{1}})^{2}(1-2q)(2-2q)}. \end{aligned} \end{aligned}$$
Proof
Using Lemma 2.3, the modulus property, Hölder’s inequality and the harmonic convexity of \(|{\Upsilon }'|^{q}\), we have
$$\begin{aligned} & \biggl\vert \frac{\Omega (\lambda ){\Upsilon }({{b_{2}}})+\Omega (\mu ){\Upsilon }({b_{1}})}{\lambda +\mu }\\ &\qquad {}- \frac{1}{\lambda +\mu } \biggl[{_{ (\frac{1}{{{b_{2}}}} )^{+}}I_{ \Phi }{\Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} +{_{ (\frac{1}{{b_{1}}} )^{-}}I_{\Phi }{\Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \biggr] \biggr\vert \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \biggl( \int _{0}^{ \lambda }{\Omega ^{p}({\tau })} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{p}}\\ &\qquad{}\times \biggl( \int _{0}^{\lambda } \frac{1}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2p}} \biggl\vert {\Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}}} \biggr) \biggr\vert ^{q} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{q}} \\ &\qquad {}+ \biggl( \int _{0}^{\mu }{\Omega ^{p}({\tau })} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{p}}\\ &\qquad{}\times \biggl( \int _{0}^{\mu } \frac{1}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2p}} \biggl\vert {\Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}}} \biggr) \biggr\vert ^{q} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{q}} \biggr] \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \biggl( \int _{0}^{ \lambda }{\Omega ^{p}({\tau })} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{\lambda }{\bigl((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}}\bigr)^{-2q}}\\ &\qquad{}\times \biggl(\frac{\mu +{\tau }}{\lambda +\mu } \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} +\frac{\lambda -{\tau }}{\lambda +\mu } \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} {\tau } \biggr)^{\frac{1}{q}} \\ &\qquad {}+ \biggl( \int _{0}^{\mu }{\Omega ^{p}({\tau })} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{p}} \biggl( \int _{0}^{\mu }{\bigl((\lambda +{\tau }){{b_{2}}}+( \mu -{\tau }){b_{1}}\bigr)^{-2q}}\\ &\qquad{}\times \biggl(\frac{\mu -{\tau }}{\lambda +\mu } \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} + \frac{\lambda +{\tau }}{\lambda +\mu } \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} {\tau } \biggr)^{\frac{1}{q}} \biggr]. \end{aligned}$$
After simple calculations, we obtain our required result. □
Corollary 2.19
Choosing \(\Phi ({\tau })={\tau }\) in Theorem 2.7, we have
$$\begin{aligned} & \biggl\vert \frac{\lambda {\Upsilon }({{b_{2}}})+\mu {\Upsilon }({b_{1}})}{\lambda +\mu }- \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \int _{\frac{1}{{{b_{2}}}}}^{ \frac{1}{{b_{1}}}}{\Upsilon \circ \Psi }(x) \,\mathrm{d}x \biggr\vert \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \biggl( \frac{\lambda ^{p+1}}{p+1} \biggr)^{\frac{1}{p}} \bigl(\sigma _{1} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{2} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}+ \biggl(\frac{\mu ^{p+1}}{p+1} \biggr)^{ \frac{1}{p}} \\ &\qquad {}\times \bigl(\sigma _{3} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{4} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \biggr], \end{aligned}$$
where \(\sigma _{1}\), \(\sigma _{2}\), \(\sigma _{3}\) and \(\sigma _{4}\) are already defined in Theorem 2.7.
Corollary 2.20
Taking \(\Phi ({\tau })=\frac{{\tau }^{\alpha }}{\Gamma (\alpha )}\) in Theorem 2.7, we obtain
$$\begin{aligned} &\biggl\vert \frac{\lambda ^{\alpha }{\Upsilon }({{b_{2}}})+\mu ^{\alpha }{\Upsilon }({b_{1}})}{\lambda +\mu }- \frac{({b_{1}{b_{2}}})^{\alpha }(\lambda +\mu )^{\alpha -1}\Gamma (\alpha +1)}{({{b_{2}}}-{b_{1}})^{\alpha }} \biggl[{J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }{\Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \\ &\qquad {}+{{J_{ (\frac{1}{{b_{1}}} )^{-}}^{\alpha }} { \Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \biggr]\biggr\vert \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \biggl( \frac{\lambda ^{{\alpha }p+1}}{{\alpha }p+1} \biggr)^{\frac{1}{p}} \bigl(\sigma _{1} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{2} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}+ \biggl(\frac{\mu ^{{\alpha }p+1}}{{\alpha }p+1} \biggr)^{\frac{1}{p}} \\ &\qquad {}\times \bigl(\sigma _{3} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{4} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \biggr], \end{aligned}$$
where \(\sigma _{1}\), \(\sigma _{2}\), \(\sigma _{3}\) and \(\sigma _{4}\) are already defined in Theorem 2.7.
Corollary 2.21
Choosing \(\Phi ({\tau })= \frac{{\tau }^{\frac{\alpha }{k}}}{k\Gamma _{k}(\alpha )}\) in Theorem 2.7, we obtain
$$\begin{aligned} &\biggl\vert \frac{\lambda ^{\frac{\alpha }{k}}{\Upsilon }({{b_{2}}})+\mu ^{\frac{\alpha }{k}}{\Upsilon }({b_{1}})}{\lambda +\mu }- \frac{k({b_{1}{b_{2}}})^{\frac{\alpha }{k}}(\lambda +\mu )^{\frac{\alpha }{k}-1}\Gamma _{k}(\alpha +k)}{({{b_{2}}}-{b_{1}})^{\frac{\alpha }{k}}} \biggl[{_{k}{J_{ ( \frac{1}{{{b_{2}}}} )^{+}}^{\alpha }} {\Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \\ &\qquad {}+{{_{k}J_{ (\frac{1}{{b_{1}}} )^{-}}^{ \alpha }} {\Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \biggr]\biggr\vert \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \biggl( \frac{k\lambda ^{\frac{{\alpha }p+k}{k}}}{{\alpha }p+k} \biggr)^{ \frac{1}{p}} \bigl(\sigma _{1} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{2} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}+ \biggl( \frac{k\mu ^{\frac{{\alpha }p+k}{k}}}{{{\alpha }p}+k} \biggr)^{ \frac{1}{p}} \\ &\qquad {}\times \bigl(\sigma _{3} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{4} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \biggr], \end{aligned}$$
where \(\sigma _{1}\), \(\sigma _{2}\), \(\sigma _{3}\) and \(\sigma _{4}\) are already defined in Theorem 2.7.
Theorem 2.8
Let \({\Upsilon }:[{b_{1}},{{b_{2}}}]\rightarrow \mathbb{R}\) be a function on \(({b_{1}},{{b_{2}}})\) with \({b_{1}}<{{b_{2}}}\) and \(|{\Upsilon }'|^{q}\) be an harmonic convex function with \(q\geq 1\) and \(\lambda ,\mu \in [0,\infty )\) with \(\lambda +\mu \neq0\), then
$$\begin{aligned} & \biggl\vert \frac{\Omega (\lambda ){\Upsilon }({{b_{2}}})+\Omega (\mu ){\Upsilon }({b_{1}})}{\lambda +\mu }\\ &\qquad{}- \frac{1}{\lambda +\mu } \biggl[{_{ (\frac{1}{{{b_{2}}}} )^{+}}I_{ \Phi }{\Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} +{_{ (\frac{1}{{b_{1}}} )^{-}}I_{\Phi }{\Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \biggr] \biggr\vert \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \biggl( \int _{0}^{ \lambda } \frac{\Omega ({\tau })}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}}\\ &\qquad{}\times\biggl( \int _{0}^{\lambda } \frac{\Omega ({\tau })}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \biggl(\frac{\mu +{\tau }}{\lambda +\mu } \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+ \frac{\lambda -{\tau }}{\lambda +\mu } \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} {\tau }\biggr)^{\frac{1}{q}}\\ &\qquad{}+ \biggl( \int _{0}^{\mu } \frac{\Omega ({\tau })}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}} \\ &\qquad {}\times \biggl( \int _{0}^{\mu } \frac{\Omega ({\tau })}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \biggl( \frac{\mu -{\tau }}{\lambda +\mu } \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+ \frac{\lambda +{\tau }}{\lambda +\mu } \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} {\tau } \biggr)^{\frac{1}{q}}\biggr]. \end{aligned}$$
Proof
Using Lemma 2.3, the modulus property, the power mean inequality and the harmonic convexity of of \(|{\Upsilon }'|^{q}\), we have
$$\begin{aligned} & \biggl\vert \frac{\Omega (\lambda ){\Upsilon }({{b_{2}}})+\Omega (\mu ){\Upsilon }({b_{1}})}{\lambda +\mu }\\ &\qquad{}- \frac{1}{\lambda +\mu } \biggl[{_{ (\frac{1}{{{b_{2}}}} )^{+}}I_{ \Phi }{\Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} +{_{ (\frac{1}{{b_{1}}} )^{-}}I_{\Phi }{\Upsilon \circ \Psi } \biggl( \frac{\lambda {{b_{2}}}+\mu {b_{1}}}{{b_{1}{b_{2}}}(\lambda +\mu )} \biggr)} \biggr] \biggr\vert \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \int _{0}^{\lambda } \frac{\Omega ({\tau })}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \biggl\vert {\Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}}} \biggr) \biggr\vert \,\mathrm{d} {\tau } \\ &\qquad {}+ \int _{0}^{\mu } \frac{\Omega ({\tau })}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \biggl\vert {\Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}}} \biggr) \biggr\vert \,\mathrm{d} {\tau }\biggr] \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \biggl( \int _{0}^{ \lambda } \frac{\Omega ({\tau })}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}}\\ &\qquad{}\times\biggl( \int _{0}^{\lambda } \frac{\Omega ({\tau })}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \\ &\qquad {}\times \biggl\vert {\Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}}} \biggr) \biggr\vert \,\mathrm{d} {\tau }\biggr)^{\frac{1}{q}}\\ &\qquad{}+ \biggl( \int _{0}^{ \mu } \frac{\Omega ({\tau })}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}} \\ &\qquad {}\times \biggl( \int _{0}^{\mu } \frac{\Omega ({\tau })}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \biggl\vert {\Upsilon }' \biggl( \frac{{b_{1}{b_{2}}}(\lambda +\mu )}{(\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}}} \biggr) \biggr\vert \,\mathrm{d} {\tau } \biggr)^{\frac{1}{q}}\biggr] \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \biggl[ \biggl( \int _{0}^{ \lambda } \frac{\Omega ({\tau })}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}}\\ &\qquad{}\times\biggl( \int _{0}^{\lambda } \frac{\Omega ({\tau })}{((\lambda -{\tau }){{b_{2}}}+(\mu +{\tau }){b_{1}})^{2}} \\ &\qquad {} \times \biggl(\frac{\mu +{\tau }}{\lambda +\mu } \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+ \frac{\lambda -{\tau }}{\lambda +\mu } \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} {\tau }\biggr)^{\frac{1}{q}}\\ &\qquad{}+ \biggl( \int _{0}^{\mu } \frac{\Omega ({\tau })}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}} \\ &\qquad {}\times \biggl( \int _{0}^{\mu } \frac{\Omega ({\tau })}{((\lambda +{\tau }){{b_{2}}}+(\mu -{\tau }){b_{1}})^{2}} \biggl( \frac{\mu -{\tau }}{\lambda +\mu } \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+ \frac{\lambda +{\tau }}{\lambda +\mu } \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q} \biggr) \,\mathrm{d} {\tau } \biggr)^{\frac{1}{q}}\biggr]. \end{aligned}$$
This completes the proof. □
Corollary 2.22
Choosing \(\Phi ({\tau })={\tau }\) and \(\lambda =\mu =1\) in Theorem 2.8, we have
$$\begin{aligned} & \biggl\vert \frac{ {\Upsilon }({{b_{2}}})+ {\Upsilon }({b_{1}})}{2}- \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \int _{\frac{1}{{{b_{2}}}}}^{ \frac{1}{{b_{1}}}}{\Upsilon \circ \Psi }(x) \,\mathrm{d}x \biggr\vert \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \bigl[\sigma _{5}^{1- \frac{1}{q}} \bigl(\sigma _{6} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{7} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}\\ &\qquad{}+ \sigma _{8}^{1- \frac{1}{q}} \bigl(\sigma _{9} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{10} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr], \end{aligned}$$
where
$$\begin{aligned} &\sigma _{5} :=\frac{({b_{1}}+{{b_{2}}})^{-2}}{2}{_{2}F_{1} \biggl(2,2,3, \frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\sigma _{6} :=\frac{({b_{1}}+{{b_{2}}})^{-2}}{4}{_{2}F_{1} \biggl(2,2,3, \frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} \biggr)}+ \frac{({b_{1}}+{{b_{2}}})^{-2}}{12}{_{2}F_{1} \biggl(2,3,4, \frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\sigma _{7} :=\frac{({b_{1}}+{{b_{2}}})^{-2}}{12}{_{2}F_{1} \biggl(2,2,4, \frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\sigma _{8} :=\frac{({b_{1}}+{{b_{2}}})^{-2}}{2}{_{2}F_{1} \biggl(2,2,3, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\sigma _{9} :=\frac{({b_{1}}+{{b_{2}}})^{-2}}{12}{_{2}F_{1} \biggl(2,2,4, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\sigma _{10} :=\frac{({b_{1}}+{{b_{2}}})^{-2}}{4}{_{2}F_{1} \biggl(2,2,3, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)}+ \frac{({b_{1}}+{{b_{2}}})^{-2}}{12}{_{2}F_{1} \biggl(2,3,4, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)}. \end{aligned}$$
Corollary 2.23
Taking \(\Phi ({\tau })=\frac{{\tau }^{\alpha }}{\Gamma (\alpha )}\) and \(\lambda =\mu =1\) in Theorem 2.8, we obtain
$$\begin{aligned} &\biggl\vert \frac{{\Upsilon }({{b_{2}}})+{\Upsilon }({b_{1}})}{2}- \frac{({b_{1}{b_{2}}})^{\alpha }2^{\alpha -1}\Gamma (\alpha +1)}{({{b_{2}}}-{b_{1}})^{\alpha }} \biggl[{J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }{\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} \biggr)} \\ &\qquad {}+{{J_{ (\frac{1}{{b_{1}}} )^{-}}^{\alpha }} { \Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} \biggr)} \biggr]\biggr\vert \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \bigl[\sigma _{11}^{1- \frac{1}{q}} \bigl(\sigma _{12} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{13} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}\\ &\qquad{}+ \sigma _{14}^{1- \frac{1}{q}} \bigl(\sigma _{15} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{16} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr], \end{aligned}$$
where
$$\begin{aligned} &\sigma _{11} :=\frac{({b_{1}}+{{b_{2}}})^{-2}}{\alpha +1}{_{2}F_{1} \biggl(2,\alpha +1,\alpha +2, \frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ & \begin{aligned} \sigma _{12} :={}&\frac{({b_{1}}+{{b_{2}}})^{-2}}{2(\alpha +1)}{_{2}F_{1} \biggl(2,\alpha +1,\alpha +2, \frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} \biggr)} \\ &{}+\frac{({b_{1}}+{{b_{2}}})^{-2}}{2(\alpha +1)(\alpha +2)} {_{2}F_{1} \biggl(2,\alpha +2, \alpha +3, \frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \end{aligned} \\ &\sigma _{13} := \frac{({b_{1}}+{{b_{2}}})^{-2}}{2(\alpha +1)(\alpha +2)}{_{2}F_{1} \biggl(2,\alpha +1,\alpha +3, \frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\sigma _{14} :=\frac{({b_{1}}+{{b_{2}}})^{-2}}{\alpha +1}{_{2}F_{1} \biggl(2,\alpha +1,\alpha +2, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\sigma _{15} := \frac{({b_{1}}+{{b_{2}}})^{-2}}{2(\alpha +1)(\alpha +2)}{_{2}F_{1} \biggl(2,\alpha +1,\alpha +3, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\begin{aligned} \sigma _{16}:={}&\frac{({b_{1}}+{{b_{2}}})^{-2}}{2(\alpha +1)}{_{2}F_{1} \biggl(2,\alpha +1,\alpha +2, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)} \\ &{}+\frac{({b_{1}}+{{b_{2}}})^{-2}}{2(\alpha +1)(\alpha +2)}{_{2}F_{1} \biggl(2,\alpha +2, \alpha +3, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)}. \end{aligned} \end{aligned}$$
Corollary 2.24
Choosing \(\Phi ({\tau })= \frac{{\tau }^{\frac{\alpha }{k}}}{k\Gamma _{k}(\alpha )}\) and \(\lambda =\mu =1\) in Theorem 2.8, we obtain
$$\begin{aligned} &\biggl\vert \frac{{\Upsilon }({{b_{2}}})+{\Upsilon }({b_{1}})}{2}- \frac{({b_{1}{b_{2}}})^{\frac{\alpha }{k}}2^{\frac{\alpha }{k}-1}\Gamma _{k}(\alpha +k)}{({{b_{2}}}-{b_{1}})^{\frac{\alpha }{k}}} \biggl[{_{k}J_{ (\frac{1}{{{b_{2}}}} )^{+}}^{\alpha }{ \Upsilon \circ \Psi } \biggl(\frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} \biggr)} \\ &\qquad {}+{{_{k}J_{ (\frac{1}{{b_{1}}} )^{-}}^{ \alpha }} {\Upsilon \circ \Psi } \biggl( \frac{{b_{1}}+{{b_{2}}}}{2{b_{1}{b_{2}}}} \biggr)} \biggr]\biggr\vert \\ &\quad \leq {b_{1}{b_{2}}}({{b_{2}}}-{b_{1}}) \bigl[\sigma _{16}^{ \frac{1}{p}} \bigl(\sigma _{17} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{18} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}\\ &\qquad{}+ \sigma _{19}^{ \frac{1}{p}} \bigl(\sigma _{20} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q}+\sigma _{21} \bigl\vert { \Upsilon }'({b_{1}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr], \end{aligned}$$
where
$$\begin{aligned} &\sigma _{16} :=\frac{k({b_{1}}+{{b_{2}}})^{-2}}{\alpha +k}{_{2}F_{1} \biggl(2,\alpha +k,\alpha +2k, \frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ & \begin{aligned} \sigma _{17} :={}&\frac{k({b_{1}}+{{b_{2}}})^{-2}}{2(\alpha k)}{_{2}F_{1} \biggl(2,\alpha +k,\alpha +2k, \frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} \biggr)} \\ &{}+\frac{k({b_{1}}+{{b_{2}}})^{-2}}{2(\alpha +k)(\alpha +2k)} {_{2}F_{1} \biggl(2,\alpha +2k, \alpha +3k, \frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \end{aligned} \\ &\sigma _{18} := \frac{k({b_{1}}+{{b_{2}}})^{-2}}{2(\alpha +k)(\alpha +2k)}{_{2}F_{1} \biggl(2,\alpha +k,\alpha +3k, \frac{{{b_{2}}}-{b_{1}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\sigma _{19} :=\frac{k({b_{1}}+{{b_{2}}})^{-2}}{\alpha +k}{_{2}F_{1} \biggl(2,\alpha +k,\alpha +2k, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ &\sigma _{20} := \frac{k({b_{1}}+{{b_{2}}})^{-2}}{2(\alpha +k)(\alpha +2k)}{_{2}F_{1} \biggl(2,\alpha +k,\alpha +3k, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)}, \\ & \begin{aligned} \sigma _{21}:={}&\frac{k({b_{1}}+{{b_{2}}})^{-2}}{2(\alpha +k)}{_{2}F_{1} \biggl(2,\alpha +k,\alpha +2k, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)} \\ &{}+\frac{k({b_{1}}+{{b_{2}}})^{-2}}{2(\alpha +k)(\alpha +2k)}{_{2}F_{1} \biggl(2,\alpha +2k, \alpha +3k, \frac{{b_{1}}-{{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)}. \end{aligned} \end{aligned}$$
Theorem 2.9
Let \({\Upsilon }:[{b_{1}},{{b_{2}}}]\subset (0,+\infty )\rightarrow \mathbb{R}\) be a differentiable function on \(({b_{1}},{{b_{2}}})\) with \({b_{1}}<{{b_{2}}}\). If \(\vert {\Upsilon }' \vert ^{q}\) is an harmonic convex function with \(q>1\) and \(\frac{1}{p}+\frac{1}{q}=1\), then
$$\begin{aligned} & \biggl\vert {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{1}{2\Delta (1)} \biggl[_{\frac{1}{{{b_{2}}}}^{+}}I_{ \Phi }{{\Upsilon }\circ {\Psi }} \biggl(\frac{1}{{b_{1}}} \biggr)+ _{ \frac{1}{{b_{1}}}^{-}}I_{\Phi }{{ \Upsilon }\circ {\Psi }} \biggl( \frac{1}{{{b_{2}}}} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2\Delta (1)} \biggl( \biggl( \int _{0}^{\frac{1}{2}} \bigl\vert \Delta ({\tau }) \bigr\vert ^{p} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{p}}+ \biggl( \int _{\frac{1}{2}}^{1} \bigl\vert \delta ({\tau }) \bigr\vert ^{p} \,\mathrm{d} {\tau } \biggr)^{\frac{1}{p}} \biggr) \\ &\qquad {}\times \bigl( \bigl(N_{1} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+N_{2} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}+ \bigl(N_{3} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+N_{4} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr), \end{aligned}$$
where
$$\begin{aligned}& \begin{aligned} N_{1}&:= \int _{0}^{\frac{1}{2}} \frac{(1-{\tau })}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2q}} \,\mathrm{d} { \tau } \\ &= \frac{2^{2-2q}{{b_{2}}}^{2-2q}-({b_{1}}+{{b_{2}}})^{2-2q}}{({{b_{2}}}-{b_{1}})^{2}(2-2q)2^{2-2q}}-{b_{1}} \frac{2^{1-2q}{{b_{2}}}^{1-2q}-({b_{1}}+{{b_{2}}})^{1-2q}}{({{b_{2}}}-{b_{1}})^{2}(1-2q)2^{1-2q}}, \end{aligned}\\& \begin{aligned} N_{2}&:= \int _{0}^{\frac{1}{2}} \frac{{\tau }}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2q}} \,\mathrm{d} { \tau } \\ &={{b_{2}}} \frac{2^{1-2q}{{b_{2}}}^{1-2q}-({b_{1}}+{{b_{2}}})^{1-2q}}{({{b_{2}}}-{b_{1}})^{2}(1-2q)2^{1-2q}}- \frac{2^{2-2q}{{b_{2}}}^{2-2q}-({b_{1}}+{{b_{2}}})^{2-2q}}{({{b_{2}}}-{b_{1}})^{2}(2-2q)2^{2-2q}}, \end{aligned}\\& \begin{aligned} N_{3}&:= \int _{0}^{\frac{1}{2}} \frac{{\tau }}{((1-{\tau }){\tau b_{1}}+{\tau {b_{2}}})^{2q}} \,\mathrm{d} { \tau } \\ &= \frac{({b_{1}}+{{b_{2}}})^{2-2q}-2^{2-2q}{b_{1}}^{2-2q}}{({{b_{2}}}-{b_{1}})^{2}(2-2q)2^{2-2q}}-{b_{1}} \frac{({b_{1}}+{{b_{2}}})^{1-2q}-2^{1-2q}{b_{1}}^{1-2q}}{({{b_{2}}}-{b_{1}})^{2}(1-2q)2^{1-2q}}, \end{aligned}\\& \begin{aligned} N_{4}&:= \int _{0}^{\frac{1}{2}} \frac{1-{\tau }}{((1-{\tau }){\tau b_{1}}+{\tau {b_{2}}})^{2q}} \,\mathrm{d} {\tau } \\ &={{b_{2}}} \frac{({b_{1}}+{{b_{2}}})^{1-2q}-2^{1-2q}{b_{1}}^{1-2q}}{({{b_{2}}}-{b_{1}})^{2}(1-2q)2^{1-2q}}- \frac{({b_{1}}+{{b_{2}}})^{2-2q}-2^{2-2q}{b_{1}}^{2-2q}}{({{b_{2}}}-{b_{1}})^{2}(2-2q)2^{2-2q}}, \end{aligned}\\& \begin{aligned} N_{5}&:= \int _{\frac{1}{2}}^{1} \frac{1-{\tau }}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2q}} \,\mathrm{d} { \tau } \\ &= \frac{({b_{1}}+{{b_{2}}})^{2-2q}-2^{2-2q}{b_{1}}^{2-2q}}{({{b_{2}}}-{b_{1}})^{2}(2-2q)2^{2-2q}}-{b_{1}} \frac{({b_{1}}+{{b_{2}}})^{1-2q}-2^{1-2q}{b_{1}}^{1-2q}}{({{b_{2}}}-{b_{1}})^{2}(1-2q)2^{1-2q}}, \end{aligned}\\& \begin{aligned} N_{6}&:= \int _{\frac{1}{2}}^{1} \frac{{\tau }}{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2q}} \,\mathrm{d} { \tau } \\ &={{b_{2}}} \frac{({b_{1}}+{{b_{2}}})^{1-2q}-2^{1-2q}{b_{1}}^{1-2q}}{({{b_{2}}}-{b_{1}})^{2}(1-2q)2^{1-2q}}- \frac{({b_{1}}+{{b_{2}}})^{2-2q}-2^{2-2q}{b_{1}}^{2-2q}}{({{b_{2}}}-{b_{1}})^{2}(2-2q)2^{2-2q}}, \end{aligned}\\& \begin{aligned} N_{7}&:= \int _{\frac{1}{2}}^{1} \frac{{\tau }}{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2q}} \,\mathrm{d} { \tau } \\ &= \frac{2^{2-2q}{{b_{2}}}^{2-2q}-({b_{1}}+{{b_{2}}})^{2-2q}}{({{b_{2}}}-{b_{1}})^{2}(2-2q)2^{2-2q}}-{b_{1}} \frac{2^{1-2q}{{b_{2}}}^{1-2q}-({b_{1}}+{{b_{2}}})^{1-2q}}{({{b_{2}}}-{b_{1}})^{2}(1-2q)2^{1-2q}}, \end{aligned}\\& \begin{aligned} N_{8}&:= \int _{\frac{1}{2}}^{1} \frac{1-{\tau }}{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2q}} \,\mathrm{d} { \tau } \\ &={{b_{2}}} \frac{2^{1-2q}{{b_{2}}}^{1-2q}-({b_{1}}+{{b_{2}}})^{1-2q}}{({{b_{2}}}-{b_{1}})^{2}(1-2q)2^{1-2q}}- \frac{2^{2-2q}{{b_{2}}}^{2-2q}-({b_{1}}+{{b_{2}}})^{2-2q}}{({{b_{2}}}-{b_{1}})^{2}(2-2q)2^{2-2q}}. \end{aligned} \end{aligned}$$
Also, it is easy to verify that \(N_{1}=N_{7}\), \(N_{2}=N_{8}\), \(N_{3}=N_{5}\) and \(N_{4}=N_{6}\).
Proof
By using Lemma 2.4, the property of modulus, Hölder’s inequality and the harmonic convexity of \(\vert {\Upsilon }' \vert ^{q}\), we obtain the desired result. We omit here the proof. □
Corollary 2.25
Taking \(\Phi ({\tau })={\tau }\) in Theorem 2.9, then
$$\begin{aligned} & \biggl\vert {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \int _{b_{1}}^{{b_{2}}} \frac{{\Upsilon }(x)}{x^{2}} \,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2} \biggl( \frac{1}{2^{p+1}(p+1)} \biggr)^{\frac{1}{p}} \\ &\qquad {}\times \bigl( \bigl(N_{1} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+N_{2} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}+ \bigl(N_{3} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+N_{4} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr). \end{aligned}$$
Corollary 2.26
Choosing \(\Phi ({\tau })=\frac{{\tau }^{\alpha }}{\Gamma (\alpha )}\) in Theorem 2.9, then
$$\begin{aligned} & \biggl\vert {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{\Gamma (\alpha +1)}{2} \biggl( \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \biggr)^{\alpha } \biggl[J_{ \frac{1}{{{b_{2}}}}^{+}}^{\alpha }{{\Upsilon }\circ {\Psi }} \biggl( \frac{1}{{b_{1}}} \biggr) +J_{\frac{1}{{b_{1}}}^{-}}^{\alpha }{{ \Upsilon } \circ {\Psi }} \biggl(\frac{1}{{{b_{2}}}} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2} \biggl( \frac{ (1+2^{\alpha p-1}(\alpha p-1) )}{2^{\alpha p}(\alpha p+1)} \biggr)^{\frac{1}{p}} \\ &\qquad {}\times \bigl( \bigl(N_{1} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+N_{2} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}+ \bigl(N_{3} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+N_{4} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr). \end{aligned}$$
Corollary 2.27
Taking \(\Phi ({\tau })= \frac{{\tau }^{\frac{\alpha }{k}}}{k\Gamma _{k}(\alpha )}\) in Theorem 2.9, then
$$\begin{aligned} & \biggl\vert {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{\Gamma _{k}(\alpha +k)}{2} \biggl( \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \biggr)^{\frac{\alpha }{k}} \biggl[ _{k}J_{\frac{1}{{{b_{2}}}}^{+}}^{\alpha }{{\Upsilon }\circ { \Psi }} \biggl(\frac{1}{{b_{1}}} \biggr) + _{k}J_{\frac{1}{{b_{1}}}^{-}}^{ \alpha }{{ \Upsilon }\circ {\Psi }} \biggl(\frac{1}{{{b_{2}}}} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2} \biggl( \frac{ (k+2^{\frac{\alpha p}{k}-1}(\alpha p-k) )}{2^{\frac{\alpha p}{k}}(\alpha p+k)} \biggr)^{\frac{1}{p}} \\ &\qquad {}\times \bigl( \bigl(N_{1} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+N_{2} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}+ \bigl(N_{3} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+N_{4} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr). \end{aligned}$$
Theorem 2.10
Let \({\Upsilon }:[{b_{1}},{{b_{2}}}]\subset (0,+\infty )\rightarrow \mathbb{R}\) be a differentiable function on \(({b_{1}},{{b_{2}}})\) with \({b_{1}}<{{b_{2}}}\). If \(\vert {\Upsilon }' \vert ^{q}\) is an harmonic convex function with \(q\geq 1\), then
$$\begin{aligned} & \biggl\vert {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{1}{2\Delta (1)} \biggl[_{\frac{1}{{{b_{2}}}}^{+}}I_{ \Phi }{{\Upsilon }\circ {\Psi }} \biggl(\frac{1}{{b_{1}}} \biggr)+ _{ \frac{1}{{b_{1}}}^{-}}I_{\Phi }{{ \Upsilon }\circ {\Psi }} \biggl( \frac{1}{{{b_{2}}}} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2\Delta (1)} \\ &\qquad {}\times \biggl[ \biggl( \int _{0}^{\frac{1}{2}} \bigl\vert \Delta ({\tau }) \bigr\vert \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}} \biggl\{ \biggl( \int _{0}^{ \frac{1}{2}} \frac{(1-{\tau }) \vert \Delta ({\tau }) \vert }{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}} \,\mathrm{d} {\tau } \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}\\ &\qquad{}+ \int _{0}^{ \frac{1}{2}} \frac{{\tau } \vert \Delta ({\tau }) \vert }{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}} \,\mathrm{d} {\tau } \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr)^{ \frac{1}{q}} \\ &\qquad {}+ \biggl( \int _{0}^{\frac{1}{2}} \frac{{\tau } \vert \Delta ({\tau }) \vert }{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}} \,\mathrm{d} {\tau } \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+ \int _{0}^{ \frac{1}{2}} \frac{(1-{\tau }) \vert \Delta ({\tau }) \vert }{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}} \,\mathrm{d} {\tau } \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr)^{ \frac{1}{q}}\biggr\} \\ &\qquad {}+ \biggl( \int _{\frac{1}{2}}^{1} \bigl\vert \delta ({\tau }) \bigr\vert \,\mathrm{d} {\tau } \biggr)^{1-\frac{1}{q}}\biggl\{ \biggl( \int _{ \frac{1}{2}}^{1} \frac{(1-{\tau }) \vert \delta ({\tau }) \vert }{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}} \,\mathrm{d} {\tau } \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}\\ &\qquad{} + \int _{ \frac{1}{2}}^{1} \frac{{\tau } \vert \delta ({\tau }) \vert }{({\tau b_{1}}+(1-{\tau }){{b_{2}}})^{2}} \,\mathrm{d} {\tau } \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr)^{ \frac{1}{q}} \\ &\qquad {} + \biggl( \int _{\frac{1}{2}}^{1} \frac{{\tau } \vert \delta ({\tau }) \vert }{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}} \,\mathrm{d} {\tau } \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}\\ &\qquad{}+ \int _{ \frac{1}{2}}^{1} \frac{(1-{\tau }) \vert \delta ({\tau }) \vert }{((1-{\tau }){b_{1}}+{\tau {b_{2}}})^{2}} \,\mathrm{d} {\tau } \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \biggr)^{ \frac{1}{q}}\biggr\} \biggr]. \end{aligned}$$
Proof
By using Lemma 2.4, the property of modulus, the power mean inequality and the convexity of \(\vert {\Upsilon }' \vert ^{q}\) we obtain the desired result. We omit here the proof. □
Corollary 2.28
Taking \(\Phi ({\tau })={\tau }\) in Theorem 2.10, then
$$\begin{aligned} & \biggl\vert {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \int _{b_{1}}^{{b_{2}}} \frac{{\Upsilon }(x)}{x^{2}} \,\mathrm{d}x \biggr\vert \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2} \biggl(\frac{1}{8} \biggr)^{1- \frac{1}{q}} \\ &\qquad {}\times \bigl[ \bigl(M_{1} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+M_{2} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}+ \bigl(M_{3} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+M_{4} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \\ &\qquad {}+ \bigl(M_{2} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q} +M_{1} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}+ \bigl(M_{4} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+M_{3} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr], \end{aligned}$$
where
$$\begin{aligned}& M_{1}:=\frac{{{b_{2}}}^{-2}}{8}{_{2}F_{1}} \biggl(2,2,3, \frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} \biggr)- \frac{{{b_{2}}}^{-2}}{24}{_{2}F_{1}} \biggl(2,3,4, \frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} \biggr), \\& M_{2}:=\frac{{{b_{2}}}^{-2}}{24}{_{2}F_{1}} \biggl(2,3,4, \frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} \biggr),\qquad M_{3}:= \frac{{b_{1}}^{-2}}{24}{_{2}F_{1}} \biggl(2,3,4, \frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} \biggr), \\& M_{4}:=\frac{{b_{1}}^{-2}}{8}{_{2}F_{1}} \biggl(2,2,3, \frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} \biggr)-\frac{{b_{1}}^{-2}}{24}{_{2}F_{1}} \biggl(2,3,4, \frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} \biggr). \end{aligned}$$
Corollary 2.29
Choosing \(\Phi ({\tau })=\frac{{\tau }^{\alpha }}{\Gamma (\alpha )}\) in Theorem 2.10, then
$$\begin{aligned} & \biggl\vert {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{\Gamma (\alpha +1)}{2} \biggl( \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \biggr)^{\alpha } \biggl[J_{ \frac{1}{{{b_{2}}}}^{+}}^{\alpha }{{\Upsilon }\circ {\Psi }} \biggl( \frac{1}{{b_{1}}} \biggr) +J_{\frac{1}{{b_{1}}}^{-}}^{\alpha }{{ \Upsilon } \circ {\Psi }} \biggl(\frac{1}{{{b_{2}}}} \biggr) \biggr] \biggr\vert \\ &\quad \leq \frac{{b_{1}{b_{2}}}({{b_{2}}}-{b_{1}})}{2} \\ &\qquad {}\times \biggl[ \biggl(\frac{1}{2^{\alpha +1}(\alpha +1)} \biggr)^{1- \frac{1}{q}} \bigl\{ \bigl(M_{5} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+M_{6} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}\\ &\qquad{}+ \bigl(M_{7} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+M_{8} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr\} \\ &\qquad {}+ \biggl( \frac{2^{\alpha }(\alpha -1)+1}{2^{\alpha +1}(\alpha +1)} \biggr)^{1- \frac{1}{q}} \bigl\{ \bigl(M_{6} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q} +M_{5} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}}\\ &\qquad{}+ \bigl(M_{8} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+M_{7} \bigl\vert { \Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr\} \biggr], \end{aligned}$$
where
$$\begin{aligned}& \begin{aligned} M_{5}:={}&\frac{{{b_{2}}}^{-2}}{2^{\alpha +1}(\alpha +1)}{_{2}F_{1}} \biggl(2,\alpha +1,\alpha +2, \frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} \biggr)\\ &{}-\frac{{{b_{2}}}^{-2}}{2^{\alpha +2}(\alpha +2)}{_{2}F_{1}} \biggl(2,\alpha +2,\alpha +3, \frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} \biggr), \end{aligned} \\& M_{6}:=\frac{{{b_{2}}}^{-2}}{2^{\alpha +2}(\alpha +2)}{_{2}F_{1}} \biggl(2,\alpha +2,\alpha +3, \frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} \biggr), \\& M_{7}:=\frac{{b_{1}}^{-2}}{2^{\alpha +2}(\alpha +2)}{_{2}F_{1}} \biggl(2, \alpha +2,\alpha +3, \frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} \biggr), \\& \begin{aligned} M_{8}:={}&\frac{{b_{1}}^{-2}}{2^{\alpha +1}(\alpha +1)}{_{2}F_{1}} \biggl(2, \alpha +1,\alpha +2, \frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} \biggr)\\ &{}- \frac{{b_{1}}^{-2}}{2^{\alpha +2}(\alpha +2)}{_{2}F_{1}} \biggl(2, \alpha +2, \alpha +3, \frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} \biggr). \end{aligned} \end{aligned}$$
Corollary 2.30
Taking \(\Phi ({\tau })= \frac{{\tau }^{\frac{\alpha }{k}}}{k\Gamma _{k}(\alpha )}\) in Theorem 2.10, then
$$\begin{aligned} & \biggl\vert {\Upsilon } \biggl(\frac{2{b_{1}{b_{2}}}}{{b_{1}}+{{b_{2}}}} \biggr)- \frac{\Gamma _{k}(\alpha +k)}{2} \biggl( \frac{{b_{1}{b_{2}}}}{{{b_{2}}}-{b_{1}}} \biggr)^{\frac{\alpha }{k}} \biggl[_{k}J_{\frac{1}{{{b_{2}}}}^{+}}^{\alpha }{{\Upsilon }\circ {\Psi }} \biggl(\frac{1}{{b_{1}}} \biggr) + _{k}J_{\frac{1}{{b_{1}}}^{-}}^{ \alpha }{{ \Upsilon }\circ {\Psi }} \biggl(\frac{1}{{{b_{2}}}} \biggr) \biggr] \biggr\vert \\ &\qquad {}\times \biggl[ \biggl(\frac{k}{2^{\frac{\alpha }{k}+1}(\alpha +k)} \biggr)^{1-\frac{1}{q}} \bigl\{ \bigl(M_{9} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+M_{10} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{ \frac{1}{q}}\\ &\qquad{}+ \bigl(M_{11} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+M_{12} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr\} \\ &\qquad {}+ \biggl( \frac{2^{\frac{\alpha }{k}}(\alpha -k)+k}{2^{\frac{\alpha }{k}+1}(\alpha +k)} \biggr)^{1-\frac{1}{q}} \bigl\{ \bigl(M_{10} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q} +M_{9} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{ \frac{1}{q}}\\ &\qquad{}+ \bigl(M_{12} \bigl\vert {\Upsilon }'({b_{1}}) \bigr\vert ^{q}+M_{11} \bigl\vert {\Upsilon }'({{b_{2}}}) \bigr\vert ^{q} \bigr)^{\frac{1}{q}} \bigr\} \biggr], \end{aligned}$$
where
$$\begin{aligned}& \begin{aligned} M_{9}:={}&\frac{k{{b_{2}}}^{-2}}{2^{\frac{\alpha }{k}+1}(\alpha +k)}{_{2}F_{1}} \biggl(2,\alpha +k,\alpha +2k, \frac{1}{k} \biggl( \frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} \biggr) \biggr) \\ &{}-\frac{k{{b_{2}}}^{-2}}{2^{\frac{\alpha }{k}+2}(\alpha +2k)}{_{2}F_{1}} \biggl(2,\alpha +2k, \alpha +3k, \frac{1}{k} \biggl( \frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} \biggr) \biggr), \end{aligned}\\& M_{10}:=\frac{k{{b_{2}}}^{-2}}{2^{\frac{\alpha }{k}+2}(\alpha +2k)}{_{2}F_{1}} \biggl(2,\alpha +2k,\alpha +3k, \frac{1}{k} \biggl( \frac{{{b_{2}}}-{b_{1}}}{2{{b_{2}}}} \biggr) \biggr), \\& M_{11}:=\frac{k{b_{1}}^{-2}}{2^{\frac{\alpha }{k}+2}(\alpha +2k)}{_{2}F_{1}} \biggl(2,\alpha +2k,\alpha +3k, \frac{1}{k} \biggl( \frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} \biggr) \biggr),\\& \begin{aligned} M_{12}:={}&\frac{k{b_{1}}^{-2}}{2^{\frac{\alpha }{k}+1}(\alpha +k)}{_{2}F_{1}} \biggl(2,\alpha +k,\alpha +2k, \frac{1}{k} \biggl( \frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} \biggr) \biggr) \\ &{}-\frac{k{b_{1}}^{-2}}{2^{\frac{\alpha }{k}+2}(\alpha +2k)}{_{2}F_{1}} \biggl(2,\alpha +2k, \alpha +3k, \frac{1}{k} \biggl( \frac{{b_{1}}-{{b_{2}}}}{2{b_{1}}} \biggr) \biggr). \end{aligned} \end{aligned}$$
Remark
For other suitable choices of function Φ, several new interesting inequalities can be found from our results. We omit here their proofs and the details are left to the interested reader.