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Hermite–Hadamard and Hermite–Hadamard–Fejer type inequalities for p-convex functions via conformable fractional integrals

Abstract

In this paper, we obtain the Hermite–Hadamard and Hermite–Hadamard–Fejer type inequalities for p-convex functions via conformable fractional integrals. We also discuss some special cases.

Introduction and preliminaries

A function \(\varUpsilon :\mathcal{W}\rightarrow \mathbb{R}\) on an interval of real line, for all \(w_{1},w_{2}\in \mathcal{W}\) and \(\kappa \in [0,1]\), is called convex if the following inequality holds:

$$ \varUpsilon \bigl(\kappa w_{1}+(1-\kappa )w_{2}\bigr)\leq \kappa \varUpsilon (w_{1})+(1- \kappa ) \varUpsilon (w_{2}). $$
(1)

Due to the importance of convex functions, many authors have given results not only for convex functions but also for their generalizations. The Hermite–Hadamard inequality [9] on a real interval was defined by

$$ \varUpsilon \biggl(\frac{w_{1}+w_{2}}{2} \biggr)\leq \frac{1}{w_{2}-w_{1}} \int ^{w_{2}}_{w_{1}}\varUpsilon (u)\,du\leq \frac{\varUpsilon (w_{1})+\varUpsilon (w_{2})}{2} $$
(2)

for all \(w_{1},w_{2}\in \mathcal{W}\) with \(w_{1}< w_{2}\). Then Fejér [8] proved the following inequality:

$$\begin{aligned} \varUpsilon \biggl(\frac{w_{1}+w_{2}}{2} \biggr) \int _{w_{1}}^{w_{2}} \curlyvee (u)\,du \leq& \frac{1}{w_{2}-w_{1}} \int ^{w_{2}}_{w_{1}} \varUpsilon (u)\curlyvee (u)\,du \\ \leq& \frac{\varUpsilon (w_{1})+\varUpsilon (w_{2})}{2} \int _{w_{1}}^{w_{2}} \curlyvee (u)\,du, \end{aligned}$$
(3)

where \(\curlyvee :[w_{1},w_{2}]\rightarrow \mathbb{R}\) is nonnegative, integrable, and symmetric to \((w_{1}+w_{2})/2\), called Hermite–Hadamard–Fejér inequality. Inequalities (2) and (3) have been further generalized in different ways not only for classical integral but also for other generalized integrals such as Riemann–Liouville fractional integral, Katugampola, ψ-Riemann–Liouville, and conformable fractional integrals etc. For more results and details see [1, 47, 1723, 2630].

Definition 1.1

([11, 12])

Suppose an interval \(\mathcal{W}\subset (0, \infty )=\mathbb{R}_{+}\) and \(p\in \mathbb{R}\setminus \{0\}\). Then a function \(\varUpsilon :\mathcal{W}\rightarrow \mathbb{R}\) is called p-convex if

$$ \varUpsilon \bigl(\bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p}\bigr]^{\frac{1}{p}} \bigr)\leq \kappa \varUpsilon (w_{1})+(1-\kappa )\varUpsilon (w_{2}) $$
(4)

holds for all \(w_{1},w_{2}\in \mathcal{W}\) and \(\kappa \in [0,1]\). If inequality (4) is in opposite order, then ϒ is called p-concave function.

Definition 1.2

([14])

Let \(\varUpsilon \in L[w_{1},w_{2}]\). The left- and right-sided Riemann–Liouville fractional integrals \(J^{\alpha }_{w_{1}+}\varUpsilon \) and \(J^{\alpha }_{w_{2}-}\varUpsilon \) of order \(\alpha \in \mathbb{C}\) with \(\mathbb{R}(\alpha )>0\) and \(w_{2} > w_{1}\geq 0\) are given by

$$ J^{\alpha }_{w_{1}+}\varUpsilon (u)=\frac{1}{\varGamma (\alpha )} \int _{w_{1}}^{u}(u-v)^{ \alpha -1}\varUpsilon (v)\,dv,\quad u>w_{1}, $$

and

$$ J^{\alpha }_{w_{2}-}\varUpsilon (u)=\frac{1}{\varGamma (\alpha )} \int _{u}^{w_{2}}(v-u)^{ \alpha -1}\varUpsilon (v)\,dv,\quad u< w_{2}, $$

respectively, where \(\varGamma (\cdot )\) is the gamma function.

Abdeljawad [2] defined the conformable fractional integral as follows.

Definition 1.3

([2])

Let \(\alpha \in (n,n+1]\) and \(\gamma =\alpha -n\). Then the left- and right-sided conformable fractional integrals of order \(\alpha >0\) are given by

$$ J_{\alpha }^{w_{1}}\varUpsilon (u)=\frac{1}{n!} \int _{w_{1}}^{u}(u-v)^{n}(v-w_{1})^{ \gamma -1} \varUpsilon (v)\,dv, $$

and

$$ ^{w_{2}}J_{\alpha }\varUpsilon (u)=\frac{1}{n!} \int _{u}^{w_{2}}(v-u)^{n}(w_{2}-v)^{ \gamma -1} \varUpsilon (v)\,dv, $$

respectively.

Note that for \(\alpha =n+1\) then \(\gamma =1\), where \(n=0,1,2,\ldots \) , and in this case conformable fractional integrals become Riemann–Liouville fractional integrals.

The classical beta function and hypergeometric function are defined, respectively, by

$$ \beta (w_{1},w_{2})= \int _{0}^{1}u^{w_{1}-1}(1-u)^{w_{2}-1} \,du $$

and

$$ {}_{2}F_{1}(w_{1},w_{2};u;v)= \frac{1}{\beta (w_{2},u-w_{2})} \int _{0}^{1}u^{w_{2}-1}(1-u)^{u-w_{2}-1}(1-vu)^{-w_{1}} \,du, $$

with \(u>w_{2}>0\), \(|v|<1\).

The incomplete beta function is defined as follows:

$$ \beta _{u}(w_{1},w_{2})= \int _{0}^{u}v^{w_{1}-1}(1-v)^{w_{2}-1}\,dv,\quad u \in [0,1]. $$

The relationship between the classical beta function and the incomplete beta function is given as follows:

$$ \beta (w_{1},w_{2})=\beta _{u}(w_{1},w_{2})+ \beta _{1-u}(w_{1},w_{2}). $$

Hermite–Hadamard type inequalities

In this section we prove some Hermite–Hadamard type inequalities for p-convex functions via conformable fractional integral.

Theorem 2.1

Let\(\varUpsilon :[w_{1},w_{2}]\subset (0,\infty )\rightarrow \mathbb{R}\)be ap-convex function such that\(\varUpsilon \in L[w_{1},w_{2}]\)and\(\alpha >0\). Then

  1. (i)

    for\(p>0\), we have

    $$\begin{aligned} \begin{aligned}[b] &\varUpsilon \biggl( \biggl[ \frac{w_{1}^{p}+w_{2}^{p}}{2} \biggr]^{1/p} \biggr) \\ &\quad \leq \frac{\varGamma (\alpha +1)}{2\varGamma (\alpha -n)(w_{2}^{p}-w_{1}^{p})^{\alpha }} \bigl[J^{w_{1}^{p}}_{\alpha }(\varUpsilon \circ \phi ) \bigl(w_{2}^{p}\bigr)+{}^{w_{2}^{p}}J_{ \alpha }( \varUpsilon \circ \phi ) \bigl(w_{1}^{p}\bigr) \bigr] \\ &\quad \leq \frac{\varUpsilon (w_{1}^{p})+\varUpsilon (w_{2}^{p})}{2}, \end{aligned} \end{aligned}$$
    (5)

    here\(\phi (u)=u^{\frac{1}{p}}\)for all\(u\in [w_{1}^{p},w_{2}^{p}]\);

  2. (ii)

    for\(p<0\), we have

    $$\begin{aligned} \begin{aligned}[b] &\varUpsilon \biggl( \biggl[ \frac{w_{1}^{p}+w_{2}^{p}}{2} \biggr]^{1/p} \biggr) \\ &\quad \leq \frac{\varGamma (\alpha +1)}{2\varGamma (\alpha -n)(w_{1}^{p}-w_{2}^{p})^{\alpha }} \bigl[{}^{w_{1}^{p}}J_{\alpha }(\varUpsilon \circ \phi ) \bigl(w_{2}^{p}\bigr)+ J^{w_{2}^{p}}_{ \alpha }( \varUpsilon \circ \phi ) \bigl(w_{1}^{p}\bigr) \bigr] \\ &\quad \leq \frac{\varUpsilon (w_{1}^{p})+\varUpsilon (w_{2}^{p})}{2}, \end{aligned} \end{aligned}$$
    (6)

    here\(\phi (u)=u^{\frac{1}{p}}\)for all\(u\in [w_{2}^{p},w_{1}^{p}]\).

Proof

(i) Since ϒ is a p-convex function on \([w_{1},w_{2}]\), we have

$$ \varUpsilon \biggl( \biggl[\frac{x^{p}+y^{p}}{2} \biggr]^{\frac{1}{p}} \biggr) \leq \frac{\varUpsilon (x)+\varUpsilon (y)}{2}. $$

Taking \(x^{p}=\kappa w_{1}^{p}+(1-\kappa )w_{2}^{p}\) and \(y^{p}=(1-\kappa )w_{1}^{p}+\kappa w_{2}^{p}\) with \(\kappa \in [0,1]\), we get

$$ \varUpsilon \biggl( \biggl[\frac{w_{1}^{p}+w_{2}^{p}}{2} \biggr]^{ \frac{1}{p}} \biggr)\leq \frac{\varUpsilon ( [\kappa w_{1}^{p}+(1-\kappa )w_{2}^{p} ]^{\frac{1}{p}} ) +\varUpsilon ( [(1-\kappa )w_{1}^{p}+\kappa w_{2}^{p} ]^{\frac{1}{p}} )}{2}. $$
(7)

Multiplying (7) by \(\frac{1}{n!}\kappa ^{n}(1-\kappa )^{\alpha -n-1}\), with \(\kappa \in (0,1)\), \(\alpha >0\), on both sides and then integrating about κ over \([0,1]\), we find

$$\begin{aligned} \begin{aligned}[b] &\frac{2}{n!}\varUpsilon \biggl( \biggl[\frac{w_{1}^{p}+w_{2}^{p}}{2} \biggr]^{\frac{1}{p}} \biggr) \int _{0}^{1}\kappa ^{n}(1-\kappa )^{ \alpha -n-1}\,d\kappa \\ &\quad \leq \frac{1}{n!} \int _{0}^{1}\kappa ^{n}(1-\kappa )^{\alpha -n-1} \varUpsilon \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{ \frac{1}{p}} \bigr)\,d\kappa \\ &\qquad {} +\frac{1}{n!} \int _{0}^{1}\kappa ^{n}(1-\kappa )^{\alpha -n-1} \varUpsilon \bigl( \bigl[(1-\kappa )w_{1}^{p}+ \kappa w_{2}^{p} \bigr]^{ \frac{1}{p}} \bigr)\,d\kappa \\ &\quad =I_{1}+I_{2}. \end{aligned} \end{aligned}$$
(8)

By setting \(u=\kappa w_{1}^{p}+(1-\kappa )w_{2}^{p}\), we have

$$\begin{aligned} \begin{aligned}[b] I_{1}&=\frac{1}{n!} \int _{0}^{1}\kappa ^{n}(1-\kappa )^{\alpha -n-1} \varUpsilon \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{ \frac{1}{p}} \bigr)\,d\kappa \\ &=\frac{1}{n!} \int _{w_{2}^{p}}^{w_{1}^{p}} \biggl( \frac{u-w_{2}^{p}}{w_{1}^{p}-w_{2}^{p}} \biggr)^{n} \biggl(1- \frac{u-w_{2}^{p}}{w_{1}^{p}-w_{2}^{p}} \biggr)^{\alpha -n-1}( \varUpsilon \circ \phi ) (u)\frac{du}{w_{1}^{p}-w_{2}^{p}} \\ &=\frac{1}{n! (w_{2}^{p}-w_{1}^{p} )^{\alpha }} \int _{w_{1}^{p}}^{w_{2}^{p}} \bigl(w_{2}^{p}-u \bigr)^{n} \bigl(u-w_{1}^{p} \bigr)^{\alpha -n-1}( \varUpsilon \circ \phi ) (u)\,du \\ &=\frac{1}{(w_{2}^{p}-w_{1}^{p})^{\alpha }}\, J^{w_{1}^{p}}_{\alpha }( \varUpsilon \circ \phi ) \bigl(w_{2}^{p}\bigr). \end{aligned} \end{aligned}$$
(9)

Similarly, by setting \(u=\kappa w_{2}^{p}+(1-\kappa )w_{1}^{p}\), we have

$$\begin{aligned} \begin{aligned}[b] I_{2}&=\frac{1}{n!} \int _{0}^{1}\kappa ^{n}(1-\kappa )^{\alpha -n-1} \varUpsilon \bigl( \bigl[\kappa w_{2}^{p}+(1- \kappa )w_{1}^{p} \bigr]^{ \frac{1}{p}} \bigr)\,d\kappa \\ &=\frac{1}{n!} \int _{w_{1}^{p}}^{w_{2}^{p}} \biggl( \frac{u-w_{1}^{p}}{w_{2}^{p}-w_{1}^{p}} \biggr)^{n} \biggl(1- \frac{u-w_{1}^{p}}{w_{2}^{p}-w_{1}^{p}} \biggr)^{\alpha -n-1}( \varUpsilon \circ \phi ) (u)\frac{du}{w_{2}^{p}-w_{1}^{p}} \\ &=\frac{1}{n! (w_{2}^{p}-w_{1}^{p} )^{\alpha }} \int _{w_{1}^{p}}^{w_{2}^{p}} \bigl(u-w_{1}^{p} \bigr)^{n} \bigl(w_{2}^{p}-u \bigr)^{\alpha -n-1}( \varUpsilon \circ \phi ) (u)\,du \\ &=\frac{1}{(w_{2}^{p}-w_{1}^{p})^{\alpha }}{}^{w_{2}^{p}}J_{\alpha }( \varUpsilon \circ \phi ) \bigl(w_{1}^{p}\bigr). \end{aligned} \end{aligned}$$
(10)

Thus, by putting values of \(I_{1}\) and \(I_{2}\) in (8), the first inequality of (5) is achieved. For another inequality, we note that

$$ \varUpsilon \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{ \frac{1}{p}} \bigr)+\varUpsilon \bigl( \bigl[\kappa w_{2}^{p}+(1- \kappa )w_{1}^{p} \bigr]^{\frac{1}{p}} \bigr) \leq \bigl[ \varUpsilon (w_{1})+ \varUpsilon (w_{2}) \bigr]. $$
(11)

Multiplying (11) by \(\frac{1}{n!}\kappa ^{n}(1-\kappa )^{\alpha -n-1}\), with \(\kappa \in (0,1)\), \(\alpha >0\), on both sides and then integrating about κ over \([0,1]\), we achieve the second inequality of (5). This completes the proof.

(ii) Proof is identical to that of (i). □

Remark 2.1

In Theorem 2.1:

  1. 1.

    If we let \(p=1\) in (i), we get Theorem 2.1 in [25].

  2. 2.

    If we let \(p=-1\) in (ii), we get Theorem 2.1 in [3].

  3. 3.

    If we let \(p=1\) and \(\alpha =n+1\) in (i), we get Theorem 2 in [24].

  4. 4.

    If we let \(p=-1\) and \(\alpha =n+1\) in (ii), we get Theorem 4 in [13].

Lemma 2.1

Let\(\varUpsilon :[w_{1},w_{2}]\subset (0,\infty )\rightarrow \mathbb{R}\)be a differentiable function on\((w_{1},w_{2})\)with\(w_{1}< w_{2}\)such that\(\varUpsilon '\in L[w_{1},w_{2}]\)and\(\alpha >0\). Then

  1. (i)

    for\(p>0\), we have

    $$\begin{aligned} \begin{aligned}[b] &{}_{1}\Delta _{\varUpsilon }(w_{1},w_{2};\alpha ;\beta ;J) \\ &\quad =\frac{w_{2}^{p}-w_{1}^{p}}{2p} \int _{0}^{1} \bigl(\beta _{1-\kappa }(n+1, \alpha -n)-\beta _{\kappa }(n+1,\alpha -n) \bigr) \\ &\qquad {}\times A_{\kappa }^{ \frac{1}{p}-1} \varUpsilon ' \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa , \end{aligned} \end{aligned}$$
    (12)

    here\(A_{\kappa }= [\kappa w_{1}^{p}+(1-\kappa )w_{2}^{p} ]\)and

    $$\begin{aligned} \begin{aligned} &{}_{1}\Delta _{\varUpsilon }(w_{1},w_{2}; \alpha ;\beta ;J) \\ &\quad =\beta (n+1,\alpha -n) \biggl( \frac{\varUpsilon (w_{1})+\varUpsilon (w_{2})}{2} \biggr) \\ &\qquad {}- \frac{n!}{2(w_{2}^{p}-w_{1}^{p})^{\alpha }} \bigl[J^{w_{1}^{p}}_{ \alpha }(\varUpsilon \circ \phi ) \bigl(w_{2}^{p}\bigr)+{}^{w_{2}^{p}}J_{\alpha }( \varUpsilon \circ \phi ) \tbinom{1}{p} \bigr]; \end{aligned} \end{aligned}$$
  2. (ii)

    for\(p<0\), we have

    $$\begin{aligned} \begin{aligned}[b] &{}_{2}\Delta _{\varUpsilon }(w_{1},w_{2};\alpha ;\beta ;J) \\ &\quad =\frac{w_{1}^{p}-w_{2}^{p}}{2p} \int _{0}^{1} \bigl(\beta _{\kappa }(n+1, \alpha -n)-\beta _{1-\kappa }(n+1,\alpha -n) \bigr) \\ &\qquad {}\times B_{\kappa }^{ \frac{1}{p}-1} \varUpsilon ' \bigl( \bigl[\kappa w_{2}^{p}+(1- \kappa )w_{1}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa , \end{aligned} \end{aligned}$$
    (13)

    here\(B_{\kappa }= [\kappa w_{2}^{p}+(1-\kappa )w_{1}^{p} ]\)and

    $$\begin{aligned} \begin{aligned} &{}_{2}\Delta _{\varUpsilon }(w_{1},w_{2}; \alpha ;\beta ;J) \\ &\quad =\beta (n+1,\alpha -n) \biggl( \frac{\varUpsilon (w_{1})+\varUpsilon (w_{2})}{2} \biggr) \\ &\qquad {} - \frac{n!}{2(w_{1}^{p}-w_{2}^{p})^{\alpha }} \bigl[{}^{w_{1}^{p}}J_{ \alpha }(\varUpsilon \circ \phi ) \bigl(w_{2}^{p}\bigr)+ J^{w_{2}^{p}}_{\alpha }( \varUpsilon \circ \phi ) \bigl(w_{1}^{p}\bigr) \bigr]. \end{aligned} \end{aligned}$$

Proof

(i) Consider

$$\begin{aligned} \begin{aligned}[b] &\int _{0}^{1} \bigl(\beta _{1-\kappa }(n+1, \alpha -n)-\beta _{ \kappa }(n+1,\alpha -n) \bigr) A_{\kappa }^{\frac{1}{p}-1} \varUpsilon ' \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{ \frac{1}{p}} \bigr)\,d\kappa \\ &\quad = \int _{0}^{1}\beta _{1-\kappa }(n+1,\alpha -n)A_{\kappa }^{ \frac{1}{p}-1} \varUpsilon ' \bigl( \bigl[ \kappa w_{1}^{p}+(1-\kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \\ &\qquad {} - \int _{0}^{1}\beta _{\kappa }(n+1,\alpha -n) A_{\kappa }^{ \frac{1}{p}-1} \varUpsilon ' \bigl( \bigl[ \kappa w_{1}^{p}+(1-\kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \\ &\quad =I_{1}-I_{2}. \end{aligned} \end{aligned}$$
(14)

Then, by integration by parts, we have

$$\begin{aligned} \begin{aligned}[b] I_{1}&= \int _{0}^{1}\beta _{1-\kappa }(n+1,\alpha -n)A_{\kappa }^{ \frac{1}{p}-1} \varUpsilon ' \bigl( \bigl[ \kappa w_{1}^{p}+(1-\kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \\ &= \int _{0}^{1} \biggl( \int _{0}^{1-\kappa }u^{n}(1-u)^{\alpha -n-1} \,du \biggr)A_{\kappa }^{\frac{1}{p}-1} \varUpsilon ' \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \\ &=\frac{p}{w_{2}^{p}-w_{1}^{p}}\beta (n+1,\alpha -n)\varUpsilon (w_{2}) \\ &\quad {} -\frac{p}{w_{2}^{p}-w_{1}^{p}} \int _{0}^{1}(1-\kappa )^{n}\kappa ^{ \alpha -n-1} \varUpsilon \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \\ &=\frac{p}{w_{2}^{p}-w_{1}^{p}}\beta (n+1,\alpha -n)\varUpsilon (w_{2}) \\ &\quad {} -\frac{p}{w_{2}^{p}-w_{1}^{p}} \int _{w_{2}^{p}}^{w_{1}^{p}} \biggl(1- \frac{x-w_{2}^{p}}{w_{1}^{p}-w_{2}^{p}} \biggr)^{n} \biggl( \frac{x-w_{2}^{p}}{w_{1}^{p}-w_{2}^{p}} \biggr)^{\alpha -n-1} \frac{(\varUpsilon \circ \phi )(x)}{w_{1}^{p}-w_{2}^{p}}dx \\ &=\frac{p}{w_{2}^{p}-w_{1}^{p}}\beta (n+1,\alpha -n)\varUpsilon (w_{2})- \frac{n!}{(w_{2}^{p}-w_{1}^{p})^{\alpha +1}}{}^{w_{2}^{p}}J_{\alpha }( \varUpsilon \circ \phi ) \bigl(w_{1}^{p}\bigr). \end{aligned} \end{aligned}$$
(15)

Similarly, we have

$$\begin{aligned} \begin{aligned}[b] I_{2}&= \int _{0}^{1}\beta _{\kappa }(n+1,\alpha -n)\varUpsilon ' \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \\ &= \int _{0}^{1} \biggl( \int _{0}^{\kappa }u^{n}(1-u)^{\alpha -n-1} \,du \biggr)\varUpsilon ' \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \\ &=-\frac{p}{w_{2}^{p}-w_{1}^{p}}\beta (n+1,\alpha -n)\varUpsilon (w_{1}) \\ &\quad {} +\frac{p}{w_{2}^{p}-w_{1}^{p}} \int _{0}^{1}\kappa ^{n}(1-\kappa )^{ \alpha -n-1} \varUpsilon \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \\ &=-\frac{p}{w_{2}^{p}-w_{1}^{p}}\beta (n+1,\alpha -n)\varUpsilon (w_{1}) \\ &\quad {} +\frac{p}{w_{2}^{p}-w_{1}^{p}} \int _{w_{2}^{p}}^{w_{1}^{p}} \biggl( \frac{x-w_{2}^{p}}{w_{1}^{p}-w_{2}^{p}} \biggr)^{n} \biggl(1- \frac{x-w_{2}^{p}}{w_{1}^{p}-w_{2}^{p}} \biggr)^{\alpha -n-1} \frac{(\varUpsilon \circ \phi )(x)}{w_{1}^{p}-w_{2}^{p}}dx \\ &=-\frac{p}{w_{2}^{p}-w_{1}^{p}}\beta (n+1,\alpha -n)\varUpsilon (w_{1})+ \frac{n!}{(w_{2}^{p}-w_{1}^{p})^{\alpha +1}}\, J^{w_{1}^{p}}_{\alpha }( \varUpsilon \circ \phi ) \bigl(w_{2}^{p}\bigr). \end{aligned} \end{aligned}$$
(16)

By substituting values of \(I_{1}\) and \(I_{2}\) in (14) and then multiplying by \(\frac{w_{2}^{p}-w_{1}^{p}}{2}\), we get (12).

(ii) Proof is similar to that of (i). □

Remark 2.2

By taking \(p=-1\) in Lemma 2.1, we obtain Lemma 2.1 in [3].

Theorem 2.2

Let\(\varUpsilon :[w_{1},w_{2}]\subset (0,\infty )\rightarrow \mathbb{R}\)be a differentiable function on\((w_{1},w_{2})\)with\(w_{1}< w_{2}\)such that\(\varUpsilon '\in L[w_{1},w_{2}]\)and\(\alpha >0\). If\(|\varUpsilon '|^{q}\), where\(q\geq 1\), is ap-convex function, then

  1. (i)

    for\(p>0\), we have

    $$ \bigl\vert {}_{1}\Delta _{\varUpsilon }(w_{1},w_{2}; \alpha ;\beta ;J) \bigr\vert \leq \frac{w_{2}^{p}-w_{1}^{p}}{2p}\lambda ^{1-1/q} \bigl(\lambda _{1} \bigl\vert \varUpsilon '(w_{1}) \bigr\vert ^{q}+\lambda _{2} \bigl\vert \varUpsilon '(w_{2}) \bigr\vert ^{q} \bigr)^{1/q}, $$
    (17)

    here

    $$\begin{aligned}& \lambda =\beta (n+1,\alpha -n+1)-\beta (n+1,\alpha -n)+\beta (n+2, \alpha -n), \\& \lambda _{1}=\frac{w_{2}^{1-p}}{2}\,{}_{2}F_{1} \biggl(1-\frac{1}{p},2;3;1-\frac{w_{1}^{p}}{w_{2}^{p}} \biggr) \quad \textit{and}\quad \lambda _{2}=\frac{w_{2}^{1-p}}{2}\,{}_{2}F_{1} \biggl(1-\frac{1}{p},1;3;1-\frac{w_{1}^{p}}{w_{2}^{p}} \biggr); \end{aligned}$$
  2. (ii)

    for\(p<0\), we have

    $$ \bigl\vert {}_{2}\Delta _{\varUpsilon }(w_{1},w_{2}; \alpha ;\beta ;J) \bigr\vert \leq \frac{w_{1}^{p}-w_{2}^{p}}{2p}\lambda _{3}^{1-1/q} \bigl(\lambda _{4} \bigl\vert \varUpsilon '(w_{1}) \bigr\vert ^{q}+ \lambda _{5} \bigl\vert \varUpsilon '(w_{2}) \bigr\vert ^{q} \bigr)^{1/q}, $$
    (18)

    here

    $$\begin{aligned}& \lambda _{3}=\beta (n+1,\alpha -n+1)-\beta (n+2,\alpha -n), \\& \lambda _{4}=\frac{w_{2}^{p-1}}{2}\,{}_{2}F_{1} \biggl(1-\frac{1}{p},1;3;1-\frac{w_{2}^{p}}{w_{1}^{p}} \biggr)\quad \textit{and}\quad \lambda _{5}=\frac{w_{2}^{p-1}}{2}\,{}_{2}F_{1} \biggl(1-\frac{1}{p},2;3;1-\frac{w_{2}^{p}}{w_{1}^{p}} \biggr). \end{aligned}$$

Proof

(i) Let \(A_{\kappa }= [\kappa w_{1}^{p}+(1-\kappa )w_{2}^{p} ]\). Applying Lemma 2.1, power mean inequality, and p-convexity of \(|\varUpsilon '|^{q}\), we find

$$\begin{aligned} \begin{aligned}[b] & \bigl\vert {}_{1}\Delta _{\varUpsilon }(w_{1},w_{2};\alpha ;\beta ;J) \bigr\vert \\ &\quad = \biggl\vert \frac{w_{2}^{p}-w_{1}^{p}}{2p} \int _{0}^{1} \bigl\{ \beta _{1- \kappa }(n+1,\alpha -n)-\beta _{\kappa }(n+1,\alpha -n) \bigr\} \\ &\qquad {}\times A_{ \kappa }^{\frac{1}{p}-1}\varUpsilon ' \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \biggr\vert \\ &\quad \leq \frac{w_{2}^{p}-w_{1}^{p}}{2p} \biggl( \int _{0}^{1} \bigl\{ \beta _{1-\kappa }(n+1,\alpha -n)-\beta _{\kappa }(n+1,\alpha -n) \bigr\} \,d\kappa \biggr)^{1-1/q} \\ &\qquad {} \times \biggl( \int _{0}^{1}A_{\kappa }^{\frac{1}{p}-1} \bigl\vert \varUpsilon ' \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{ \frac{1}{p}} \bigr) \bigr\vert ^{q}\,d\kappa \biggr)^{1/q} \\ &\quad \leq \frac{w_{2}^{p}-w_{1}^{p}}{2p}\lambda ^{1-1/q} \biggl( \int _{0}^{1}A_{ \kappa }^{\frac{1}{p}-1} \bigl[\kappa \bigl\vert \varUpsilon '(w_{1}) \bigr\vert ^{q}+(1- \kappa ) \bigl\vert \varUpsilon '(w_{2}) \bigr\vert ^{q} \bigr]\,d\kappa \biggr)^{1/q} \\ &\quad =\frac{w_{2}^{p}-w_{1}^{p}}{2p}\lambda ^{1-1/q} \bigl(\lambda _{1} \bigl\vert \varUpsilon '(w_{1}) \bigr\vert ^{q}+\lambda _{2} \bigl\vert \varUpsilon '(w_{2}) \bigr\vert ^{q} \bigr)^{1/q}, \end{aligned} \end{aligned}$$
(19)

where

$$\begin{aligned}& \begin{aligned} \lambda &= \int _{0}^{1} \bigl(\beta _{1-\kappa }(n+1, \alpha -n)- \beta _{\kappa }(n+1,\alpha -n) \bigr)\,d\kappa \\ &= \int _{0}^{1} \biggl( \int _{0}^{1-\kappa }u^{n}(1-u)^{\alpha -n-1} \,du \biggr)\,d\kappa + \int _{0}^{1} \biggl( \int _{0}^{\kappa }u^{n}(1-u)^{ \alpha -n-1} \,du \biggr)\,d\kappa \\ &=\kappa \biggl( \int _{0}^{1-\kappa }u^{n}(1-u)^{\alpha -n-1} \,du \biggr)\bigg|_{0}^{1}+ \int _{0}^{1}\kappa (1-\kappa )^{n} \kappa ^{ \alpha -n-1}\,d\kappa \\ &\quad {} +\kappa \biggl( \int _{0}^{\kappa }u^{n}(1-u)^{\alpha -n-1} \,du \biggr) \bigg|_{0}^{1}+ \int _{0}^{1}\kappa ^{n+1}(1-\kappa )^{\alpha -n-1}\,d\kappa \\ &=\beta (n+1,\alpha -n+1)-\beta (n+1,\alpha -n)+\beta (n+2,\alpha -n), \end{aligned} \\& \lambda _{1}= \int _{0}^{1}\kappa A_{\kappa }^{\frac{1}{p}-1} \,d\kappa = \frac{w_{2}^{1-p}}{2}\,{}_{2}F_{1} \biggl( 1- \frac{1}{p},2;3;1-\frac{w_{1}^{p}}{w_{2}^{p}} \biggr) , \end{aligned}$$

and

$$ \lambda _{2}= \int _{0}^{1}(1-\kappa )A_{\kappa }^{\frac{1}{p}-1} \,d\kappa =\frac{w_{2}^{1-p}}{2}\,{}_{2}F_{1} \biggl( 1- \frac{1}{p},1;3;1-\frac{w_{1}^{p}}{w_{2}^{p}} \biggr). $$

Hence the proof is completed.

(ii) Proof is similar to that of (i). □

Remark 2.3

By letting \(p=-1\) in Theorem 2.2, we obtain Theorem 2.2 in [3].

Now, for the next two results, we consider the case when \(p>0\) and leave the case when \(p<0\) for the reader.

Theorem 2.3

Let\(\varUpsilon :[w_{1},w_{2}]\subset (0,\infty )\rightarrow \mathbb{R}\)be a differentiable function on\((w_{1},w_{2})\)with\(w_{1}< w_{2}\)such that\(\varUpsilon '\in L[w_{1},w_{2}]\)and\(\alpha >0\). If\(|\varUpsilon '|^{q}\), where\(q\geq 1\), is ap-convex function, then for\(p>0\), we have

$$\begin{aligned} \begin{aligned}[b] & \bigl\vert {}_{1}\Delta _{\varUpsilon }(w_{1},w_{2};\alpha ;\beta ;J) \bigr\vert \\ &\quad \leq \frac{w_{2}^{p}-w_{1}^{p}}{2p}\mu ^{1-1/q} \bigl((\mu _{1}- \mu _{2}) \bigl\vert \varUpsilon '(w_{1}) \bigr\vert ^{q}+(\mu _{3}-\mu _{4}) \bigl\vert \varUpsilon '(w_{2}) \bigr\vert ^{q} \bigr)^{1/q}, \end{aligned} \end{aligned}$$
(20)

here

$$\begin{aligned}& \mu =\frac{w_{2}^{1-p}}{2}\,{}_{2}F_{1}\biggl(1- \frac{1}{p},1;2;1-\frac{w_{1}^{p}}{w_{2}^{p}}\biggr), \\& \mu _{1}=\frac{1}{2}\beta (n+1,\alpha -n+2), \\& \mu _{2}=\frac{1}{2} \bigl(\beta (n+1,\alpha -n)-\beta (n+3,\alpha -n) \bigr), \\& \mu _{3}=\beta (n+2,\alpha -n+1)-\frac{1}{2}\beta (n+1, \alpha -n+2), \end{aligned}$$

and

$$ \mu _{4}=\frac{1}{2}\beta (n+1,\alpha -n)+ \frac{1}{2}\beta (n+3, \alpha -n)-\beta (n+2,\alpha -n). $$

Proof

Applying Lemma 2.1, power mean inequality, and p-convexity of \(|\varUpsilon '|^{q}\), we have

$$\begin{aligned} \begin{aligned}[b] & \bigl\vert {}_{1}\Delta _{\varUpsilon }(w_{1},w_{2};\alpha ;\beta ;J) \bigr\vert \\ &\quad = \biggl\vert \frac{w_{2}^{p}-w_{1}^{p}}{2p} \int _{0}^{1} \bigl\{ \beta _{1- \kappa }(n+1,\alpha -n)-\beta _{\kappa }(n+1,\alpha -n) \bigr\} \\ &\qquad {}\times A_{ \kappa }^{\frac{1}{p}-1}\varUpsilon ' \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \biggr\vert \\ &\quad \leq \frac{w_{2}^{p}-w_{1}^{p}}{2p} \biggl( \int _{0}^{1}A_{\kappa }^{ \frac{1}{p}-1} \,d\kappa \biggr)^{1-1/q} \\ &\qquad {} \times \biggl( \int _{0}^{1} \bigl\{ \beta _{1-\kappa }(n+1,\alpha -n)- \beta _{\kappa }(n+1,\alpha -n) \bigr\} \bigl\vert \varUpsilon ' \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr) \bigr\vert ^{q}\,d\kappa \biggr)^{1/q} \\ &\quad \leq \frac{w_{2}^{p}-w_{1}^{p}}{2p}\mu ^{1-1/q} \biggl( \int _{0}^{1} \bigl\{ \beta _{1-\kappa }(n+1,\alpha -n)-\beta _{\kappa }(n+1, \alpha -n) \bigr\} \\ &\qquad {} \times \bigl[\kappa \bigl\vert \varUpsilon '(w_{1}) \bigr\vert ^{q}+(1-\kappa ) \bigl\vert \varUpsilon '(w_{2}) \bigr\vert ^{q} \bigr]\,d \kappa \biggr)^{1/q} \\ &\quad =\frac{w_{2}^{p}-w_{1}^{p}}{2p}\mu ^{1-1/q} \bigl((\mu _{1}-\mu _{2}) \bigl\vert \varUpsilon '(w_{1}) \bigr\vert ^{q}+(\mu _{3}-\mu _{4}) \bigl\vert \varUpsilon '(w_{2}) \bigr\vert ^{q} \bigr)^{1/q}, \end{aligned} \end{aligned}$$
(21)

where

$$\begin{aligned}& \mu = \int _{0}^{1}A_{\kappa }^{\frac{1}{p}-1} \,d\kappa = \frac{w_{2}^{1-p}}{2}\,{}_{2}F_{1}\biggl(1- \frac{1}{p},1;2;1-\frac{w_{1}^{p}}{w_{2}^{p}}\biggr), \\& \mu _{1}= \int _{0}^{1}\kappa \beta _{1-\kappa }(n+1, \alpha -n)\,d\kappa =\frac{1}{2}\beta (n+1,\alpha -n+2), \\& \mu _{2}= \int _{0}^{1}\kappa \beta _{\kappa }(n+1, \alpha -n)= \frac{1}{2} \bigl(\beta (n+1,\alpha -n)-\beta (n+3,\alpha -n) \bigr), \\& \mu _{3}= \int _{0}^{1}(1-\kappa )\beta _{1-\kappa }(n+1,\alpha -n)\,d\kappa =\beta (n+2,\alpha -n+1)- \frac{1}{2}\beta (n+1,\alpha -n+2), \end{aligned}$$

and

$$\begin{aligned} \mu _{4} =& \int _{0}^{1}(1-\kappa )\beta _{\kappa }(n+1,\alpha -n)\,d\kappa \\ =&\frac{1}{2}\beta (n+1,\alpha -n)+\frac{1}{2}\beta (n+3, \alpha -n)-\beta (n+2,\alpha -n). \end{aligned}$$

Hence the proof is completed. □

Theorem 2.4

Let\(\varUpsilon :[w_{1},w_{2}]\subset (0,\infty )\rightarrow \mathbb{R}\)be a differentiable function on\((w_{1},w_{2})\)with\(w_{1}< w_{2}\)such that\(\varUpsilon '\in L[w_{1},w_{2}]\)and\(\alpha >0\). If\(|\varUpsilon '|^{q}\), where\(q,l> 1\)with\(\frac{1}{q}+\frac{1}{l}=1\), is ap-convex function, then

$$\begin{aligned} \begin{aligned}[b] & \bigl\vert {}_{1}\Delta _{\varUpsilon }(w_{1},w_{2};\alpha ;\beta ;J) \bigr\vert \\ &\quad \leq \frac{w_{2}^{p}-w_{1}^{p}}{2p}\nu ^{\frac{1}{l}} \bigl(\nu _{1} \bigl\vert \varUpsilon '(w_{1}) \bigr\vert ^{q}+\nu _{2} \bigl\vert \varUpsilon '(w_{2}) \bigr\vert ^{q} \bigr)^{1/q}, \end{aligned} \end{aligned}$$
(22)

here

$$\begin{aligned}& \nu =2 \int _{0}^{\frac{1}{2}} \biggl( \int _{\kappa }^{1-\kappa }u^{n}(1-u)^{ \alpha -n-1} \,du \biggr)\,d\kappa , \\& \nu _{1}=\frac{w_{2}^{q(1-p)}}{2}\,{}_{2}F_{1} \biggl( q\biggl(1-\frac{1}{p}\biggr),2;3;1-\frac{w_{1}^{p}}{w_{2}^{p}} \biggr) , \\& \nu _{2}=\frac{w_{2}^{q(1-p)}}{2}\,{}_{2}F_{1} \biggl( q\biggl(1-\frac{1}{p}\biggr),1;3;1-\frac{w_{1}^{p}}{w_{2}^{p}} \biggr). \end{aligned}$$

Proof

Let \(A_{\kappa }= [\kappa w_{1}^{p}+(1-\kappa )w_{2}^{p} ]\). Applying Lemma 2.1, Hölder’s inequality, and p-convexity of \(|\varUpsilon '|^{q}\), we have

$$\begin{aligned} \begin{aligned}[b] & \bigl\vert {}_{1}\Delta _{\varUpsilon }(w_{1},w_{2};\alpha ;\beta ;J) \bigr\vert \\ &\quad = \biggl\vert \frac{w_{2}^{p}-w_{1}^{p}}{2p} \int _{0}^{1} \bigl\{ \beta _{1- \kappa }(n+1,\alpha -n)-\beta _{\kappa }(n+1,\alpha -n) \bigr\} \\ &\qquad \times{}A_{ \kappa }^{\frac{1}{p}-1}\varUpsilon ' \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \biggr\vert \\ &\quad \leq \frac{w_{2}^{p}-w_{1}^{p}}{2p} \biggl( \int _{0}^{1} \bigl\vert \beta _{1-\kappa }(n+1,\alpha -n)-\beta _{\kappa }(n+1,\alpha -n) \bigr\vert ^{l}\,d\kappa \biggr)^{\frac{1}{l}} \\ &\qquad {} \times \biggl( \int _{0}^{1}A_{\kappa }^{q(\frac{1}{p}-1)} \bigl\vert \varUpsilon ' \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{ \frac{1}{p}} \bigr) \bigr\vert ^{q}\,d\kappa \biggr)^{1/q} \\ &\quad \leq \frac{w_{2}^{p}-w_{1}^{p}}{2p}\nu ^{\frac{1}{l}} \biggl( \int _{0}^{1}A_{ \kappa }^{q(\frac{1}{p}-1)} \bigl[\kappa \bigl\vert \varUpsilon '(w_{1}) \bigr\vert ^{q}+(1- \kappa ) \bigl\vert \varUpsilon '(w_{2}) \bigr\vert ^{q} \bigr]\,d\kappa \biggr)^{1/q} \\ &\quad =\frac{w_{2}^{p}-w_{1}^{p}}{2p}\nu ^{\frac{1}{p}} \bigl(\nu _{1} \bigl\vert \varUpsilon '(w_{1}) \bigr\vert ^{q}+\nu \bigl\vert \varUpsilon '(w_{2}) \bigr\vert ^{q} \bigr)^{1/q}, \end{aligned} \end{aligned}$$
(23)

where

$$\begin{aligned}& \begin{aligned} \nu &= \int _{0}^{1} \bigl\vert \beta _{1-\kappa }(n+1,\alpha -n)-\beta _{ \kappa }(n+1,\alpha -n) \bigr\vert ^{l}\,d\kappa \\ &= \int _{0}^{\frac{1}{2}} \bigl(\beta _{1-\kappa }(n+1, \alpha -n)- \beta _{\kappa }(n+1,\alpha -n) \bigr)^{l}\,d \kappa \\ &\quad {} + \int _{\frac{1}{2}}^{1} \bigl(\beta _{\kappa }(n+1, \alpha -n)-\beta _{1- \kappa }(n+1,\alpha -n) \bigr)^{l}\,d\kappa \\ &= \int _{0}^{\frac{1}{2}} \biggl( \int _{\kappa }^{1-\kappa }u^{n}(1-u)^{ \alpha -n-1} \,du \biggr)^{l}\,d\kappa + \int _{\frac{1}{2}}^{1} \biggl( \int _{1-\kappa }^{\kappa }u^{n}(1-u)^{\alpha -n-1} \,du \biggr)^{l}\,d\kappa \\ &=2 \int _{0}^{\frac{1}{2}} \biggl( \int _{\kappa }^{1-\kappa }u^{n}(1-u)^{ \alpha -n-1} \,du \biggr)^{l}\,d\kappa , \end{aligned} \\& \nu _{1}= \int _{0}^{1}\kappa A_{\kappa }^{q(\frac{1}{p}-1)} \,d\kappa = \frac{w_{2}^{q(1-p)}}{2}\,{}_{2}F_{1} \biggl( q \biggl(1-\frac{1}{p}\biggr),2;3;1-\frac{w_{1}^{p}}{w_{2}^{p}} \biggr) , \end{aligned}$$

and

$$ \nu _{2}= \int _{0}^{1}(1-\kappa )A_{\kappa }^{q(\frac{1}{p}-1)} \,d\kappa =\frac{w_{2}^{q(1-p)}}{2}\,{}_{2}F_{1} \biggl( q \biggl(1-\frac{1}{p}\biggr),1;3;1-\frac{w_{1}^{p}}{w_{2}^{p}} \biggr) . $$

Hence the proof is completed. □

Hermite–Hadamard–Fejér type inequalities

In this section we prove some Hermite–Hadamard–Fejér type inequalities for p-convex functions via conformable fractional integral. First we give the following useful definition.

Definition 3.1

([15])

Let \(p\in \mathbb{R}\setminus \{0\}\). A function \(\curlyvee :[w_{1},w_{2}]\subseteq (0,\infty )\rightarrow \mathbb{R} \) is called p-symmetric around \([\frac{w_{1}^{p}+w_{2}^{p}}{2} ]^{1/p}\) if

$$ \curlyvee (x)=\curlyvee \bigl( \bigl[w_{1}^{p}+w_{2}^{p}-x^{p} \bigr]^{\frac{1}{p}} \bigr) $$

holds for all \(x\in [w_{1},w_{2}]\).

Now we prove the following identity.

Lemma 3.1

Let\(p\in \mathbb{R}\setminus \{0\}\). If\(\curlyvee :[w_{1},w_{2}]\subseteq (0,\infty )\rightarrow \mathbb{R} \)is integrable andp-symmetric around\([\frac{w_{1}^{p}+w_{2}^{p}}{2} ]^{1/p}\), then

  1. (i)

    for\(p>0\), we have

    $$ \begin{aligned}[b] J_{\alpha }^{w_{1}^{p}}(\curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr)&={}^{w_{2}^{p}}J_{ \alpha }( \curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr) \\ &= \frac{1}{2} \bigl[J_{ \alpha }^{w_{1}^{p}}(\curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr)+{}^{w_{2}^{p}}J_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr) \bigr], \end{aligned} $$
    (24)

    with\(\alpha >0\)and\(\phi (u)=u^{\frac{1}{p}}\), for all\(u\in [w_{1}^{p},w_{2}^{p}]\);

  2. (ii)

    for\(p<0\), we have

    $$ \begin{aligned}[b] J_{\alpha }^{w_{2}^{p}}(\curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr)&={}^{w_{1}^{p}}J_{ \alpha }( \curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr) \\ &= \frac{1}{2} \bigl[J_{ \alpha }^{w_{2}^{p}}(\curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr)+{}^{w_{1}^{p}}J_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr) \bigr], \end{aligned} $$
    (25)

    with\(\alpha >0\)and\(\phi (u)=u^{\frac{1}{p}}\), for all\(u\in [w_{2}^{p},w_{1}^{p}]\).

Proof

(i) Since is p-symmetric around \([\frac{w_{1}^{p}+w_{2}^{p}}{2} ]^{1/p}\), then by definition we have \(\curlyvee (x^{\frac{1}{p}})=\curlyvee ( [w_{1}^{p}+w_{2}^{p}-x ]^{\frac{1}{p}} )\) for all \(x\in [w_{1}^{p},w_{2}^{p}]\). In the following integral, setting \(u=w_{1}^{p}+w_{2}^{p}-x\) gives

$$\begin{aligned} \begin{aligned}[b] J^{w_{1}^{p}}_{\alpha }(\curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr)&= \frac{1}{n!} \int _{w_{1}^{p}}^{w_{2}^{p}} \bigl(w_{2}^{p}-u \bigr)^{n} \bigl(u-w_{1}^{p} \bigr)^{\alpha -n-1}\curlyvee \bigl(u^{\frac{1}{p}}\bigr)\,du \\ &=\frac{1}{n!} \int _{w_{1}^{p}}^{w_{2}^{p}} \bigl(x-w_{1}^{p} \bigr)^{n} \bigl(w_{2}^{p}-x \bigr)^{\alpha -n-1}\curlyvee \bigl( \bigl[w_{1}^{p}+w_{2}^{p}-x \bigr]^{\frac{1}{p}} \bigr)dx \\ &=\frac{1}{n!} \int _{w_{1}^{p}}^{w_{2}^{p}} \bigl(x-w_{1}^{p} \bigr)^{n} \bigl(w_{2}^{p}-x \bigr)^{\alpha -n-1}\curlyvee \bigl(x^{ \frac{1}{p}} \bigr)dx \\ &={}^{w_{2}^{p}}J_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr). \end{aligned} \end{aligned}$$
(26)

This completes the proof.

(ii) Proof is similar to that of (i). □

Remark 3.1

In Lemma 3.1:

  1. 1.

    By taking \(\alpha =n+1\), we obtain Lemma 1 in [16].

  2. 2.

    By taking \(\alpha =n+1\) and \(p=1\), we find Lemma 3 of [10].

Corollary 3.1

Under the assumptions of Lemma 3.1:

  1. 1.

    If\(p=1\)in (i), then we get

    $$ J_{\alpha }^{w_{1}}\curlyvee (w_{2})={}^{w_{2}}J_{\alpha } \curlyvee (w_{1}) =\frac{1}{2} \bigl[J_{\alpha }^{w_{1}} \curlyvee (w_{2})+{}^{w_{2}}J_{ \alpha }\curlyvee (w_{1}) \bigr]. $$
    (27)
  2. 2.

    If\(p=-1\)in (ii), then we get

    $$\begin{aligned} \begin{aligned}[b] {}^{1/w_{1}}J_{\alpha }(\curlyvee \circ \phi ) (1/w_{2})&=J_{\alpha }^{1/w_{2}}( \curlyvee \circ \phi ) (1/w_{1}) \\ &=\frac{1}{2} \bigl[{}^{1/w_{1}}J_{\alpha }(\curlyvee \circ \phi ) (1/w_{2})+ J^{1/w_{2}}_{\alpha }(\curlyvee \circ \phi ) (1/w_{1}) \bigr]. \end{aligned} \end{aligned}$$
    (28)

Theorem 3.2

Let\(p\in \mathbb{R}\setminus \{0\}\). Let\(\varUpsilon :[w_{1},w_{2}]\subset (0,\infty )\rightarrow \mathbb{R}\)be ap-convex function with\(w_{1}< w_{2}\)and\(\varUpsilon \in L[w_{1},w_{2}]\). If\(\curlyvee :[w_{1},w_{2}]\subseteq \mathbb{R}\setminus \{0\} \rightarrow \mathbb{R}\)is nonnegative, integrable, andp-symmetric around\([\frac{w_{1}^{p}+w_{2}^{p}}{2} ]^{1/p}\). Then

  1. (i)

    for\(p>0\), the following inequalities hold:

    $$\begin{aligned} \begin{aligned}[b] &\varUpsilon \biggl( \biggl[ \frac{w_{1}^{p}+w_{2}^{p}}{2} \biggr]^{1/p} \biggr) \bigl[J^{w_{1}^{p}}_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr)+ {}^{w_{2}^{p}}J_{\alpha }(\curlyvee \circ \phi ) \bigl(w_{1}^{p} \bigr) \bigr] \\ &\quad \leq \bigl[J^{w_{1}^{p}}_{\alpha }\bigl(\varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{2}^{p}\bigr)+ {}^{w_{2}^{p}}J_{\alpha } \bigl(\varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{1}^{p} \bigr) \bigr] \\ &\quad \leq \frac{\varUpsilon (w_{1})+\varUpsilon (w_{2})}{2} \bigl[J^{w_{1}^{p}}_{ \alpha }(\curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr)+{}^{w_{2}^{p}}J_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr) \bigr], \end{aligned} \end{aligned}$$
    (29)

    with\(\alpha >0\)and\(\phi (x)=x^{\frac{1}{p}}\), for all\(x\in [w_{1}^{p},w_{2}^{p}]\);

  2. (ii)

    for\(p<0\), the following inequalities hold:

    $$\begin{aligned} \begin{aligned}[b] &\varUpsilon \biggl( \biggl[ \frac{w_{1}^{p}+w_{2}^{p}}{2} \biggr]^{1/p} \biggr) \bigl[J^{w_{2}^{p}}_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr)+ {}^{w_{1}^{p}}J_{\alpha }(\curlyvee \circ \phi ) \bigl(w_{2}^{p} \bigr) \bigr] \\ &\quad \leq \bigl[J^{w_{2}^{p}}_{\alpha }\bigl(\varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{1}^{p}\bigr)+ {}^{w_{1}^{p}}J_{\alpha } \bigl(\varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{2}^{p} \bigr) \bigr] \\ &\quad \leq \frac{\varUpsilon (w_{1})+\varUpsilon (w_{2})}{2} \bigl[J^{w_{2}^{p}}_{ \alpha }(\curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr)+{}^{w_{1}^{p}}J_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr) \bigr], \end{aligned} \end{aligned}$$
    (30)

    with\(\alpha >0\)and\(\phi (x)=x^{\frac{1}{p}}\), for all\(x\in [w_{2}^{p},w_{1}^{p}]\).

Proof

(i) Since ϒ is a p-convex function on \([w_{1},w_{2}]\), we have

$$ \varUpsilon \biggl( \biggl[\frac{x^{p}+y^{p}}{2} \biggr]^{\frac{1}{p}} \biggr) \leq \frac{\varUpsilon (x)+\varUpsilon (y)}{2}. $$

Taking \(x^{p}=\kappa w_{1}^{p}+(1-\kappa )w_{2}^{p}\) and \(y^{p}=(1-\kappa )w_{1}^{p}+\kappa w_{2}^{p}\) with \(\kappa \in [0,1]\), we get

$$ \varUpsilon \biggl( \biggl[\frac{w_{1}^{p}+w_{2}^{p}}{2} \biggr]^{ \frac{1}{p}} \biggr)\leq \frac{\varUpsilon ( [\kappa w_{1}^{p}+(1-\kappa )w_{2}^{p} ]^{\frac{1}{p}} ) +\varUpsilon ( [(1-\kappa )w_{1}^{p}+\kappa w_{2}^{p} ]^{\frac{1}{p}} )}{2}. $$
(31)

Multiplying (31) by \(\frac{1}{n!}\kappa ^{n}(1-\kappa )^{\alpha -n-1}\curlyvee ( [\kappa w_{1}^{p}+(1-\kappa )w_{2}^{p} ]^{\frac{1}{p}} )\) on both sides, \(\alpha >0\) and then integrating about κ over \([0,1]\), we obtain

$$\begin{aligned} \begin{aligned}[b] &\frac{2}{n!}\varUpsilon \biggl( \biggl[ \frac{w_{1}^{p}+w_{2}^{p}}{2} \biggr]^{\frac{1}{p}} \biggr) \int _{0}^{1}\kappa ^{n}(1-\kappa )^{ \alpha -n-1} \curlyvee \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \\ &\quad \leq \frac{1}{n!} \int _{0}^{1}\kappa ^{n}(1-\kappa )^{\alpha -n-1} \varUpsilon \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{ \frac{1}{p}} \bigr) \curlyvee \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \\ &\qquad {} +\frac{1}{n!} \int _{0}^{1}\kappa ^{n}(1-\kappa )^{\alpha -n-1} \varUpsilon \bigl( \bigl[(1-\kappa )w_{1}^{p}+ \kappa w_{2}^{p} \bigr]^{ \frac{1}{p}} \bigr) \curlyvee \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa . \end{aligned} \end{aligned}$$
(32)

Since is nonnegative, integrable, and p-symmetric with respect to \([\frac{w_{1}^{p}+w_{2}^{p}}{2} ]^{1/p}\), then

$$ \curlyvee \bigl( \bigl[\kappa w_{1}^{p}+(1-\kappa )w_{2}^{p} \bigr]^{ \frac{1}{p}} \bigr)= \curlyvee \bigl( \bigl[\kappa w_{2}^{p}+(1- \kappa )w_{1}^{p} \bigr]^{\frac{1}{p}} \bigr). $$

Also choosing \(u=\kappa w_{1}^{p}+(1-\kappa )w_{2}^{p}\) leads to

$$\begin{aligned} &\frac{2}{n! (w_{2}^{p}-w_{1}^{p} )^{\alpha }}\varUpsilon \biggl( \biggl[ \frac{w_{1}^{p}+w_{2}^{p}}{2} \biggr]^{\frac{1}{p}} \biggr) \int _{w_{1}^{p}}^{w_{2}^{p}} \bigl(w_{2}^{p}-u \bigr)^{n} \bigl(u-w_{1}^{p} \bigr)^{\alpha -n-1} \curlyvee \bigl(u^{\frac{1}{p}}\bigr)\,du \\ &\quad \leq \frac{1}{n! (w_{2}^{p}-w_{1}^{p} )^{\alpha }} \biggl[ \int _{w_{1}^{p}}^{w_{2}^{p}} \bigl(w_{2}^{p}-u \bigr)^{n} \bigl(u-w_{1}^{p} \bigr)^{\alpha -n-1} \varUpsilon \bigl(u^{\frac{1}{p}}\bigr)\curlyvee \bigl(u^{ \frac{1}{p}}\bigr)\,du \\ &\qquad {} + \int _{w_{1}^{p}}^{w_{2}^{p}} \bigl(w_{2}^{p}-u \bigr)^{n} \bigl(u-w_{1}^{p} \bigr)^{\alpha -n-1} \varUpsilon \bigl( \bigl[w_{1}^{p}+w_{2}^{p}-u \bigr]^{\frac{1}{p}} \bigr)\curlyvee \bigl(u^{\frac{1}{p}}\bigr)\,du \biggr] \\ &\quad =\frac{1}{n! (w_{2}^{p}-w_{1}^{p} )^{\alpha }} \biggl[ \int _{w_{1}^{p}}^{w_{2}^{p}} \bigl(w_{2}^{p}-u \bigr)^{n} \bigl(u-w_{1}^{p} \bigr)^{\alpha -n-1} \varUpsilon \bigl(u^{\frac{1}{p}}\bigr)\curlyvee \bigl(u^{\frac{1}{p}}\bigr)\,du \\ &\qquad {} + \int _{w_{1}^{p}}^{w_{2}^{p}} \bigl(u-w_{1}^{p} \bigr)^{n} \bigl(w_{2}^{p}-u \bigr)^{\alpha -n-1} \varUpsilon \bigl(u^{\frac{1}{p}}\bigr)\curlyvee \bigl( \bigl[w_{1}^{p}+w_{2}^{p}-u \bigr]^{\frac{1}{p}} \bigr)\,du \biggr]. \end{aligned}$$
(33)

Therefore, by Lemma 3.1 we have

$$\begin{aligned} \begin{aligned}[b] &\frac{1}{ (w_{2}^{p}-w_{1}^{p} )^{\alpha }}\varUpsilon \biggl( \biggl[\frac{w_{1}^{p}+w_{2}^{p}}{2} \biggr]^{\frac{1}{p}} \biggr) \bigl[J^{w_{1}^{p}}_{\alpha }(\curlyvee \circ \phi ) \bigl(b^{p}\bigr)+ {}^{w_{2}^{p}}J_{\alpha }(\curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr) \bigr] \\ &\quad \leq \frac{1}{ (w_{2}^{p}-w_{1}^{p} )^{\alpha }} \bigl[J^{w_{1}^{p}}_{ \alpha }\bigl( \varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{2}^{p} \bigr)+{}^{w_{2}^{p}}J_{ \alpha }\bigl(\varUpsilon (\curlyvee \circ \phi ) \bigr) \bigl(w_{1}^{p}\bigr) \bigr]. \end{aligned} \end{aligned}$$
(34)

This completes the first inequality of (29). For the second inequality, we first note that if ϒ is a p-convex function, then we have

$$ \varUpsilon \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{ \frac{1}{p}} \bigr)+\varUpsilon \bigl( \bigl[\kappa w_{2}^{p}+(1- \kappa )w_{1}^{p} \bigr]^{\frac{1}{p}} \bigr) \leq \bigl[ \varUpsilon (w_{1})+ \varUpsilon (w_{2}) \bigr]. $$
(35)

Multiplying (35) by \(\frac{1}{n!}\kappa ^{n}(1-\kappa )^{\alpha -n-1}\curlyvee ( [\kappa w_{1}^{p}+(1-\kappa )w_{2}^{p} ]^{\frac{1}{p}} )\) on both sides, \(\alpha >0\) and then integrating about κ over \([0,1]\), we obtain

$$\begin{aligned} \begin{aligned}[b] &\frac{1}{n!} \int _{0}^{1}\kappa ^{n}(1-\kappa )^{\alpha -n-1} \varUpsilon \bigl( \bigl[\kappa _{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{ \frac{1}{p}} \bigr) \curlyvee \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \\ &\qquad {} +\frac{1}{n!} \int _{0}^{1}\kappa ^{n}(1-\kappa )^{\alpha -n-1} \varUpsilon \bigl( \bigl[\kappa w_{2}^{p}+(1- \kappa )w_{1}^{p} \bigr]^{ \frac{1}{p}} \bigr) \curlyvee \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa \\ &\quad \leq \bigl[\varUpsilon (w_{1})+\varUpsilon (w_{2}) \bigr] \frac{1}{n!} \int _{0}^{1}\kappa ^{n}(1-\kappa )^{\alpha -n-1}\curlyvee \bigl( \bigl[\kappa w_{1}^{p}+(1- \kappa )w_{2}^{p} \bigr]^{\frac{1}{p}} \bigr)\,d\kappa. \end{aligned} \end{aligned}$$
(36)

That is,

$$\begin{aligned} \begin{aligned}[b] &\frac{1}{ (w_{2}^{p}-w_{1}^{p} )^{\alpha }} \bigl[J^{w_{1}^{p}}_{ \alpha } \bigl(\varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{2}^{p} \bigr)+{}^{w_{2}^{p}}J_{ \alpha }\bigl(\varUpsilon (\curlyvee \circ \phi ) \bigr) \bigl(w_{1}^{p}\bigr) \bigr] \\ &\quad \leq \frac{1}{ (w_{2}^{p}-w_{1}^{p} )^{\alpha }} \biggl[ \frac{\varUpsilon (w_{1})+\varUpsilon (w_{2})}{2} \biggr] \bigl[J^{w_{1}^{p}}_{ \alpha }(\curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr)+{}^{w_{2}^{p}}J_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr) \bigr]. \end{aligned} \end{aligned}$$
(37)

This completes the proof.

(ii) Proof is similar to that of (i). □

Remark 3.2

In Theorem 3.2:

  1. 1.

    If \(\alpha =n+1\), we obtain Theorem 9 in [16].

  2. 2.

    If \(\alpha =n+1\) and \(p=1\), we find Theorem 4 in [10].

Corollary 3.3

Under the assumptions of Theorem 3.2:

  1. 1.

    If\(p=1\), then

    $$\begin{aligned} \begin{aligned}[b] &\varUpsilon \biggl(\frac{w_{1}+w_{2}}{2} \biggr) \bigl[J^{w_{1}}_{ \alpha }\curlyvee (w_{2})+{}^{w_{2}}J_{\alpha } \curlyvee (w_{1}) \bigr] \\ &\quad \leq \bigl[J^{w_{1}}_{\alpha }(\varUpsilon \curlyvee ) (w_{2})+{}^{w_{2}}J_{ \alpha }(\varUpsilon \curlyvee ) (w_{1}) \bigr] \\ &\quad \leq \frac{\varUpsilon (w_{1})+\varUpsilon (w_{2})}{2} \bigl[J^{w_{1}}_{ \alpha }\curlyvee (w_{2})+{}^{w_{2}}J_{\alpha }\curlyvee (w_{1}) \bigr]. \end{aligned} \end{aligned}$$
    (38)
  2. 2.

    If\(p=-1\), then

    $$\begin{aligned} \begin{aligned}[b] &\varUpsilon \biggl(\frac{2w_{1}w_{2}}{w_{1}+w_{2}} \biggr) \bigl[{}^{1/w_{1}}J_{ \alpha }(\curlyvee \circ \phi ) (1/w_{2})+J^{1/w_{2}}_{\alpha }( \curlyvee \circ \phi ) (1/w_{1}) \bigr] \\ &\quad \leq \bigl[{}^{1/w_{1}}J_{\alpha }\bigl(\varUpsilon (\curlyvee \circ \phi )\bigr) (1/w_{2})+J^{1/w_{2}}_{ \alpha }\bigl( \varUpsilon (\curlyvee \circ \phi )\bigr) (1/w_{1}) \bigr] \\ &\quad \leq \frac{\varUpsilon (w_{1})+\varUpsilon (w_{2})}{2} \bigl[{}^{1/w_{1}}J_{ \alpha }( \curlyvee \circ \phi ) (1/w_{2})+J^{1/w_{2}}_{\alpha }( \curlyvee \circ \phi ) (1/w_{1}) \bigr]. \end{aligned} \end{aligned}$$
    (39)

Remark 3.3

In Corollary 3.3(1), if we take \(\alpha =n+1\), we get inequality (3).

Lemma 3.2

Let\(p\in \mathbb{R}\setminus \{0\}\)and\(\alpha >0\). Let\(\varUpsilon :[w_{1},w_{2}]\subset (0,\infty )\rightarrow \mathbb{R}\)be a differentiable mapping and\(\varUpsilon \in L[w_{1},w_{2}]\). If\(\curlyvee :[w_{1},w_{2}]\subseteq \mathbb{R}\setminus \{0\} \rightarrow \mathbb{R}\)is nonnegative, integrable, andp-symmetric around\([\frac{w_{1}^{p}+w_{2}^{p}}{2} ]^{1/p}\), then

  1. (i)

    for\(p>0\), the following inequality holds:

    $$\begin{aligned} \begin{aligned}[b] &\frac{\varUpsilon (w_{1})+\varUpsilon (w_{2})}{2} \bigl[J^{w_{1}^{p}}_{ \alpha }(\curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr)+{}^{w_{2}^{p}}J_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr) \bigr] \\ &\qquad {} - \bigl[J^{w_{1}^{p}}_{\alpha }\bigl(\varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{2}^{p}\bigr)+ {}^{w_{2}^{p}}J_{\alpha }\bigl(\varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{1}^{p}\bigr) \bigr] \\ &\quad \leq \frac{1}{n!} \int _{w_{1}^{p}}^{w_{2}^{p}} \biggl[ \int _{w_{1}^{p}}^{t}\bigl(w_{2}^{p}-s \bigr)^{n}\bigl(s-w_{1}^{p} \bigr)^{ \alpha -n-1}(\curlyvee \circ \phi ) (s)\,ds \\ &\qquad {} - \int _{t}^{w_{2}^{p}}\bigl(s-w_{1}^{p} \bigr)^{n}\bigl(w_{2}^{p}-s \bigr)^{\alpha -n-1}( \curlyvee \circ \phi ) (s)\,ds \biggr](\varUpsilon \circ \phi )'(t) \,dt, \end{aligned} \end{aligned}$$
    (40)

    where\(\phi (x)=x^{1/p}\)for all\(x\in [w^{p}_{1},w^{p}_{2}]\);

  2. (ii)

    for\(p<0\), the following inequality holds:

    $$\begin{aligned} \begin{aligned}[b] &\frac{\varUpsilon (w_{1})+\varUpsilon (w_{2})}{2} \bigl[J^{w_{2}^{p}}_{ \alpha }(\curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr)+{}^{w_{1}^{p}}J_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr) \bigr] \\ &\qquad {} - \bigl[J^{w_{2}^{p}}_{\alpha }\bigl(\varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{1}^{p}\bigr)+ {}^{w_{1}^{p}}J_{\alpha }\bigl(\varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{2}^{p}\bigr) \bigr] \\ &\quad \leq \frac{1}{n!} \int _{w_{2}^{p}}^{w_{1}^{p}} \biggl[ \int _{w_{2}^{p}}^{t}\bigl(w_{1}^{p}-s \bigr)^{n}\bigl(s-w_{2}^{p} \bigr)^{ \alpha -n-1}(\curlyvee \circ \phi ) (s)\,ds \\ &\qquad {} - \int _{t}^{w_{1}^{p}}\bigl(s-w_{2}^{p} \bigr)^{n}\bigl(w_{1}^{p}-s \bigr)^{\alpha -n-1}( \curlyvee \circ \phi ) (s)\,ds \biggr](\varUpsilon \circ \phi )'(t) \,dt, \end{aligned} \end{aligned}$$
    (41)

    where\(\phi (x)=x^{1/p}\)for all\(x\in [w^{p}_{2},w^{p}_{1}]\).

Proof

(i) Note that

$$\begin{aligned} \begin{aligned}[b] I&= \int _{w_{1}^{p}}^{w_{2}^{p}} \biggl( \int _{w_{1}^{p}}^{t}\bigl(w_{2}^{p}-s \bigr)^{n}\bigl(s-w_{1}^{p} \bigr)^{ \alpha -n-1}(\curlyvee \circ \phi ) (s)\,ds \biggr) (\varUpsilon \circ \phi )'(t)\,dt \\ &\quad {} - \int _{w_{1}^{p}}^{w_{2}^{p}} \biggl( \int _{t}^{w_{2}^{p}}\bigl(s-w_{1}^{p} \bigr)^{n}\bigl(w_{2}^{p}-s \bigr)^{ \alpha -n-1}(\curlyvee \circ \phi ) (s)\,ds \biggr) (\varUpsilon \circ \phi )'(t) \,dt \\ &=I_{1}-I_{2}. \end{aligned} \end{aligned}$$
(42)

Integrating by parts and using Lemma 3.1, we get

$$\begin{aligned} \begin{aligned}[b] I_{1}&= \biggl( \int _{w_{1}^{p}}^{t}\bigl(w_{2}^{p}-s \bigr)^{n}\bigl(s-w_{1}^{p} \bigr)^{ \alpha -n-1}(\curlyvee \circ \phi ) (s)\,ds \biggr) (\varUpsilon \circ \phi ) (t) \bigg|_{w_{1}^{p}}^{w_{2}^{p}} \\ &\quad {} - \int _{w_{1}^{p}}^{w_{2}^{p}}\bigl(w_{2}^{p}-t \bigr)^{n}\bigl(t-w_{1}^{p} \bigr)^{\alpha -n-1}( \curlyvee \circ \phi ) (t) (\varUpsilon \circ \phi ) (t) \,dt \\ &=n! \bigl[(\varUpsilon \circ \phi ) \bigl(w_{2}^{p} \bigr)J^{w_{1}^{p}}_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr)-J^{w_{1}^{p}}_{\alpha } \bigl(\varUpsilon ( \curlyvee \circ \phi )\bigr) \bigl(w_{2}^{p} \bigr) \bigr] \\ &=n! \biggl[\frac{(\varUpsilon \circ \phi )(w_{2}^{p})}{2} \bigl\{ {}^{w_{2}^{p}}J_{ \alpha }( \curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr)+ J^{w_{1}^{p}}_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{2}^{p} \bigr) \bigr\} -J^{w_{1}^{p}}_{\alpha }\bigl( \varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{2}^{p}\bigr) \biggr]. \end{aligned} \end{aligned}$$
(43)

Similarly,

$$\begin{aligned} \begin{aligned}[b] I_{2}&= \biggl( \int _{t}^{w_{2}^{p}}\bigl(s-w_{1}^{p} \bigr)^{n}\bigl(w_{2}^{p}-s \bigr)^{ \alpha -n-1}(\curlyvee \circ \phi ) (s)\,ds \biggr) (\varUpsilon \circ \phi ) (t) \bigg|_{w_{1}^{p}}^{w_{2}^{p}} \\ &\quad {} + \int _{w_{1}^{p}}^{w_{2}^{p}}\bigl(t-w_{1}^{p} \bigr)^{n}\bigl(w_{2}^{p}-t \bigr)^{\alpha -n-1}( \curlyvee \circ \phi ) (t) (\varUpsilon \circ \phi ) (t) \,dt \\ &=n! \bigl[-(\varUpsilon \circ \phi ) \bigl(w_{1}^{p} \bigr){}^{w_{2}^{p}}J_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr)-{}^{w_{2}^{p}}J_{\alpha } \bigl(\varUpsilon ( \curlyvee \circ \phi )\bigr) \bigl(w_{1}^{p} \bigr) \bigr] \\ &=n! \biggl[\frac{-(\varUpsilon \circ \phi )(w_{1}^{p})}{2} \bigl\{ {}^{w_{2}^{p}}J_{ \alpha }( \curlyvee \circ \phi ) \bigl(w_{1}^{p} \bigr)+J^{w_{1}^{p}}_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{2}^{p}\bigr) \bigr\} \\ &\quad {}+{}^{w_{2}^{p}}J_{\alpha } \bigl( \varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{1}^{p} \bigr) \biggr]. \end{aligned} \end{aligned}$$
(44)

Thus from (43) and (44) we get

$$\begin{aligned} \begin{aligned}[b] I&=I_{1}-I_{2} \\ &=n! \biggl[\frac{\varUpsilon (w_{1})+\varUpsilon (w_{2})}{2} \bigl[J^{w_{1}^{p}}_{ \alpha }( \curlyvee \circ \phi ) \bigl(w_{2}^{p} \bigr)+{}^{w_{2}^{p}}J_{\alpha }( \curlyvee \circ \phi ) \bigl(w_{1}^{p}\bigr) \bigr] \\ &\quad {} - \bigl[J^{w_{1}^{p}}_{\alpha }\bigl(\varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{2}^{p}\bigr)+ {}^{w_{2}^{p}}J_{\alpha }\bigl(\varUpsilon (\curlyvee \circ \phi )\bigr) \bigl(w_{1}^{p}\bigr) \bigr] \biggr]. \end{aligned} \end{aligned}$$
(45)

Multiplying (45) by \(\frac{1}{n!}\), we obtain (40).

(ii) Proof is similar to that of (i). □

Remark 3.4

In Lemma 3.2:

  1. 1.

    If we take \(\alpha =n+1\), we get Lemma 2 in [16].

  2. 2.

    If we take \(\alpha =n+1\) and \(p=1\), we get Lemma 4 in [10].

Lemma 3.2 also holds for convex functions and harmonically convex functions just by taking \(p=1\) and \(p=-1\), respectively. Also, from Lemma 3.2 we can establish more useful results.

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Mehreen, N., Anwar, M. Hermite–Hadamard and Hermite–Hadamard–Fejer type inequalities for p-convex functions via conformable fractional integrals. J Inequal Appl 2020, 107 (2020). https://doi.org/10.1186/s13660-020-02363-3

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Keywords

  • Hermite–Hadamard inequality
  • Hermite–Hadamard–Fejer inequality
  • p-convex functions
  • Conformable fractional integrals