In order to prove our main results we need a lemma, which we present in this section.
Lemma 2.1
Let
\(\alpha\in(0, 1]\), \(c_{1}, c_{2} \in\mathbb{R}\)
with
\(0\leq c_{1}< c_{2}\)
and
\(\psi:[c_{1}, c_{2}]\rightarrow\mathbb {R}\)
be an
α-fractional differentiable function on
\((c_{1}, c_{2})\). Then the identity
$$\begin{aligned} &\frac{\psi(c_{1})+\psi(c_{2})}{2}-\frac{\alpha}{c_{2}^{\alpha }-c_{1}^{\alpha}} \int_{c_{1}}^{c_{2}}\psi(\xi)\,d_{\alpha}\xi \\ &\quad =\frac{(c_{2}-c_{1})}{2(c_{2}^{\alpha}-c_{1}^{\alpha})} \biggl[ \int _{0}^{1} \bigl(\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{2\alpha-1}-c_{2}^{\alpha}\bigl( \theta c_{1}+(1-\theta)c_{2}\bigr)^{\alpha-1} \bigr) \\ &\quad\quad{} \times\mathrm{D}_{\alpha}(\psi) \bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)\theta^{1-\alpha}\,d_{\alpha}\theta \\ & \quad\quad{} + \int_{0}^{1} \bigl(\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{2\alpha-1}-c_{1}^{\alpha}\bigl( \theta c_{1}+(1-\theta)c_{2}\bigr)^{\alpha-1} \bigr) \\ &\quad\quad{} \times\mathrm{D}_{\alpha}(\psi) \bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)\theta^{1-\alpha}\,d_{\alpha}\theta \biggr] \end{aligned}$$
holds if
\(\mathrm{D}_{\alpha}(\psi)\in \mathrm{L}_{\alpha}^{1}([c_{1}, c_{2}])\).
Proof
Let \(\xi=\theta c_{1}+(1-\theta)c_{2}\). Then making use of integration by parts, we get
$$\begin{aligned} & \int_{0}^{1} \bigl(\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{2\alpha-1}-c_{2}^{\alpha}\bigl( \theta c_{1}+(1-\theta)c_{2}\bigr)^{\alpha-1} \bigr) \mathrm{D} _{\alpha}(\psi) \bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)\,d\theta \\ &\quad\quad{} + \int_{0}^{1} \bigl(\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{2\alpha-1}-c_{1}^{\alpha}\bigl( \theta c_{1}+(1-\theta)c_{2}\bigr)^{\alpha-1} \bigr) \mathrm{D} _{\alpha}(\psi) \bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)\,d\theta \\ &\quad = \int_{0}^{1} \bigl(\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{\alpha}-c_{2}^{\alpha} \bigr)\psi'\bigl(\theta c_{1}+(1-\theta)c_{2} \bigr)\,d\theta \\ &\quad\quad{} + \int_{0}^{1} \bigl(\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{\alpha}-c_{1}^{\alpha} \bigr) \psi'\bigl(\theta c_{1}+(1-\theta)c_{2}\bigr)\,d\theta \\ &\quad = \bigl(\bigl(\theta c_{1}+(1-\theta)c_{2} \bigr)^{\alpha}-c_{2}^{\alpha} \bigr)\frac{\psi(\theta c_{1}+(1-\theta)c_{2})}{c_{1}-c_{2}} \bigg| _{0}^{1} \\ &\quad\quad{} - \int_{0}^{1}\alpha\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{\alpha-1}(c_{1}-c_{2}) \frac{\psi(\theta c_{1}+(1-\theta)c_{2})}{c_{1}-c_{2}}\,d\theta \\ &\quad\quad{} + \bigl(\bigl(\theta c_{1}+(1-\theta)c_{2} \bigr)^{\alpha}-c_{1}^{\alpha} \bigr)\frac{\psi(\theta c_{1}+(1-\theta)c_{2})}{c_{1}-c_{2}} \bigg| _{0}^{1} \\ &\quad\quad{} - \int_{0}^{1}\alpha\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{\alpha-1}(c_{1}-c_{2}) \frac{\psi(\theta c_{1}+(1-\theta)c_{2})}{c_{1}-c_{2}}\,d\theta \\ &\quad =\frac{1}{c_{2}-c_{1}} \biggl[\bigl(c_{2}^{\alpha}-c_{1}^{\alpha} \bigr)\psi (c_{1})-\alpha \int_{c_{1}}^{c_{2}}\psi(\xi)\,d_{\alpha}\xi \biggr] \\ &\quad\quad{} +\frac{1}{c_{2}-c_{1}} \biggl[\bigl(c_{2}^{\alpha}-c_{1}^{\alpha} \bigr)\psi(c_{2}) -\alpha \int_{c_{1}}^{c_{2}}\psi(\xi)\,d_{\alpha}\xi \biggr] \\ &\quad =\frac{c_{2}^{\alpha}-c_{1}^{\alpha}}{c_{2}-c_{1}} \bigl(\psi(c_{1})+\psi(c_{2}) \bigr) - \frac{2\alpha}{c_{2}-c_{1}} \int_{c_{1}}^{c_{2}}\psi(\xi)\,d_{\alpha}\xi. \end{aligned}$$
(2.1)
Therefore, Lemma 2.1 follows easily from (2.1). □
Remark 2.2
We clearly see that the identity given in Lemma 2.1 reduces to the identity given in Theorem 1.1 if \(\alpha=1\).
Theorem 2.3
Let
\(\alpha\in(0, 1]\), \(c_{1}, c_{2}\in\mathbb{R}\)
with
\(0\leq c_{1} < c_{2}\)
and
\(\psi:[c_{1}, c_{2}] \rightarrow\mathbb{R}\)
be an
α-differentiable function. Then the inequality
$$\begin{aligned} & \biggl\vert \frac{\psi(c_{1})+\psi(c_{2})}{2}-\frac{\alpha }{c_{2}^{\alpha}-c_{1}^{\alpha}} \int_{c_{1}}^{c_{2}}\psi(\xi)\,d_{\alpha }\xi \biggr\vert \\ &\quad \leq\frac{c_{2}-c_{1}}{2(c_{2}^{\alpha}-c_{1}^{\alpha})} \biggl[\frac{( \vert \psi^{\prime}(c_{1}) \vert + \vert \psi ^{\prime}(c_{2}) \vert ) (5c_{2}^{\alpha}-7c_{1}^{\alpha}+c_{1}c_{2}^{\alpha -1}+c_{1}^{\alpha-1}c_{2} )}{12} \biggr] \end{aligned}$$
(2.2)
holds if
\(\mathrm{D} _{\alpha}(\psi)\in \mathrm{L}_{\alpha}^{1}([c_{1}, c_{2}])\)
and
\(\vert \psi^{\prime } \vert \)
is convex on
\([c_{1}, c_{2}]\).
Proof
It follows from Lemma 2.1 and the convexities of the functions \(\xi\rightarrow\xi^{\alpha-1}\) and \(\xi\rightarrow-\xi^{\alpha}\) on \((0, \infty)\) together with the convexity of \(\vert \psi^{\prime } \vert \) on \([c_{1}, c_{2}]\) that
$$\begin{aligned}& \biggl\vert \frac{\psi(c_{1})+\psi(c_{2})}{2}-\frac{\alpha }{c_{2}^{\alpha}-c_{1}^{\alpha}} \int_{c_{1}}^{c_{2}}\psi(\xi)\,d_{\alpha }\xi \biggr\vert \\& \quad \leq\frac{c_{2}-c_{1}}{2(c_{2}^{\alpha}-c_{1}^{\alpha})} \biggl[ \int _{0}^{1} \bigl(\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{\alpha}-c_{1}^{\alpha} \bigr) \bigl\vert \psi ^{\prime}\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr) \bigr\vert \,d\theta \\& \quad\quad{} + \int_{0}^{1} \bigl(c_{2}^{\alpha}- \bigl(\theta c_{1}+(1-\theta)c_{2}\bigr)^{\alpha} \bigr) \bigl\vert \psi^{\prime}\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr) \bigr\vert \,d\theta \biggr] \\& \quad \leq\frac{c_{2}-c_{1}}{2(c_{2}^{\alpha}-c_{1}^{\alpha})} \biggl[ \int _{0}^{1} \bigl(\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{\alpha-1}\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)-c_{1}^{\alpha} \bigr) \bigl\vert \psi^{\prime }\bigl(\theta c_{1}+(1-\theta)c_{2}\bigr) \bigr\vert \,d\theta \\& \quad\quad{} + \int_{0}^{1} \bigl(c_{2}^{\alpha}- \bigl((1-\theta)c_{1}^{\alpha}+\theta c_{2}^{\alpha} \bigr) \bigr) \bigl\vert \psi^{\prime}\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr) \bigr\vert \,d\theta \biggr] \\& \quad \leq\frac{c_{2}-c_{1}}{2(c_{2}^{\alpha}-c_{1}^{\alpha})} \biggl[ \int _{0}^{1} \bigl(\bigl((1-\theta)c_{1}^{\alpha-1}+ \theta c_{2}^{\alpha-1}\bigr) \bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)-c_{1}^{\alpha} \bigr) \bigl\vert \psi^{\prime }\bigl(\theta c_{1}+(1-\theta)c_{2}\bigr) \bigr\vert \,d\theta \\& \quad\quad{} + \int_{0}^{1} \bigl(c_{2}^{\alpha}- \bigl((1-\theta)c_{1}^{\alpha}+\theta c_{2}^{\alpha} \bigr) \bigr) \bigl\vert \psi^{\prime}\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr) \bigr\vert \,d\theta \biggr] \\& \quad \leq\frac{c_{2}-c_{1}}{2(c_{2}^{\alpha}-c_{1}^{\alpha})} \biggl[ \int _{0}^{1} \bigl(\bigl((1-\theta)c_{1}^{\alpha-1}+ \theta c_{2}^{\alpha-1}\bigr) \bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)-c_{1}^{\alpha} \bigr) \\& \quad\quad{}\times \bigl[(1- \theta) \bigl\vert \psi^{\prime}(c_{1}) \bigr\vert +\theta \bigl\vert \psi'(c_{2}) \bigr\vert \bigr]\,d\theta \\& \quad\quad{} + \int_{0}^{1} \bigl(c_{2}^{\alpha}- \bigl((1-\theta)c_{1}^{\alpha}+\theta c_{2}^{\alpha} \bigr) \bigr) \bigl[(1-\theta) \bigl\vert \psi^{\prime }(c_{1}) \bigr\vert +\theta \bigl\vert \psi'(c_{2}) \bigr\vert \bigr]\,d\theta \biggr] \\& \quad =\frac{c_{2}-c_{1}}{c_{2}^{\alpha}-c_{1}^{\alpha}} \biggl[\frac {1}{4}c_{1}^{\alpha} \bigl\vert \psi^{\prime}(c_{1}) \bigr\vert + \frac{1}{12}c_{1}^{\alpha} \bigl\vert \psi^{\prime}(c_{2}) \bigr\vert +\frac{1}{12}c_{1}c_{2}^{\alpha-1} \bigl\vert \psi^{\prime}(c_{1}) \bigr\vert + \frac{1}{12}c_{1}c_{2}^{\alpha-1} \bigl\vert \psi^{\prime}(c_{2}) \bigr\vert \\& \quad\quad{} +\frac{1}{12}c_{1}^{\alpha-1}c_{2} \bigl\vert \psi^{\prime }(c_{1}) \bigr\vert +\frac{1}{12}c_{1}^{\alpha-1}c_{2} \bigl\vert \psi^{\prime}(c_{2}) \bigr\vert +\frac{1}{12}c_{2}^{\alpha} \bigl\vert \psi^{\prime}(c_{1}) \bigr\vert + \frac{1}{4}c_{2}^{\alpha} \bigl\vert \psi^{\prime}(c_{2}) \bigr\vert \\& \quad\quad{} -\frac{1}{2}c_{1}^{\alpha} \bigl\vert \psi^{\prime}(c_{1}) \bigr\vert - \frac{1}{2}c_{1}^{\alpha} \bigl\vert \psi^{\prime}(c_{2}) \bigr\vert +\frac{1}{2}c_{2}^{\alpha} \bigl\vert \psi^{\prime}(c_{1}) \bigr\vert +\frac{1}{2}c_{2}^{\alpha} \bigl\vert \psi^{\prime}(c_{2}) \bigr\vert - \frac{1}{3}c_{1}^{\alpha} \bigl\vert \psi^{\prime}(c_{1}) \bigr\vert \\& \quad\quad{} -\frac{1}{6}c_{1}^{\alpha} \bigl\vert \psi^{\prime}(c_{2}) \bigr\vert - \frac{1}{6}c_{2}^{\alpha} \bigl\vert \psi^{\prime}(c_{1}) \bigr\vert -\frac{1}{3}c_{2}^{\alpha} \bigl\vert \psi^{\prime}(c_{2}) \bigr\vert \biggr] \\& \quad =\frac{c_{2}-c_{1}}{2(c_{2}^{\alpha}-c_{1}^{\alpha})} \biggl[\frac{( \vert \psi^{\prime}(c_{1}) \vert + \vert \psi ^{\prime}(c_{2}) \vert ) (5c_{2}^{\alpha}-7c_{1}^{\alpha}+c_{1}c_{2}^{\alpha -1}+c_{1}^{\alpha-1}c_{2} )}{12} \biggr]. \end{aligned}$$
□
Remark 2.4
Let \(\alpha=1\). Then inequality (2.2) becomes
$$ \biggl\vert \frac{\psi(c_{1})+\psi(c_{2})}{2}-\frac{1}{c_{2}-c_{1}} \int _{c_{1}}^{c_{2}}\psi(\xi)\,d\xi \biggr\vert \leq \frac{c_{2}-c_{1}}{4} \bigl[ \bigl\vert \psi^{\prime}(c_{1}) \bigr\vert + \bigl\vert \psi^{\prime}(c_{2}) \bigr\vert \bigr]. $$
Theorem 2.5
Let
\(\alpha\in(0, 1]\), \(q>1\), \(c_{1}, c_{2}\in\mathbb{R}\)
with
\(0\leq c_{1} < c_{2}\)
and
\(\psi:[c_{1}, c_{2}] \rightarrow \mathbb{R}\)
be an
α-differentiable function on
\((c_{1}, c_{2})\). Then the inequality
$$\begin{aligned} & \biggl\vert \frac{\psi(c_{1})+\psi(c_{2})}{2}-\frac{\alpha }{c_{2}^{\alpha}-c_{1}^{\alpha}} \int_{c_{1}}^{c_{2}}\psi(\xi)\,d_{\alpha }\xi \biggr\vert \\ &\quad \leq\frac{c_{2}-c_{1}}{2(c_{2}^{\alpha}-c_{1}^{\alpha})} \bigl[ \bigl(\mathrm{A}_{1}(\alpha) \bigr)^{1-1/q} \bigl\{ \mathrm {A}_{2}(\alpha) \bigl\vert \psi^{\prime}(c_{1}) \bigr\vert ^{q}+ \mathrm{A}_{3} (\alpha) \bigl\vert \psi^{\prime}(c_{2}) \bigr\vert ^{q} \bigr\} ^{1/q} \\ &\quad\quad{} + \bigl(\mathrm{B}_{1}(\alpha) \bigr)^{1-1/q} \bigl\{ \mathrm {B}_{2}(\alpha) \bigl\vert \psi^{\prime}(c_{1}) \bigr\vert ^{q} +\mathrm{B}_{3}(\alpha) \bigl\vert \psi^{\prime}(c_{2}) \bigr\vert ^{q} \bigr\} ^{1/q} \bigr] \end{aligned}$$
(2.3)
is valid if
\(\mathrm{D}_{\alpha}(\psi)\in \mathrm{L}_{\alpha}^{1}([c_{1}, c_{2}])\)
and
\(\vert \psi^{\prime } \vert ^{q}\)
is convex on
\([c_{1}, c_{2}]\), where
$$\begin{aligned}& \mathrm{A}_{1}(\alpha)= \biggl[\frac{c_{1}^{\alpha+1}-c_{2}^{\alpha +1}}{(\alpha+1)(c_{1}-c_{2})} \biggr]-c_{1}^{\alpha},\quad \quad \mathrm{B}_{1}( \alpha)=c_{2}^{\alpha}- \biggl[\frac{c_{1}^{\alpha +1}-c_{2}^{\alpha+1}}{(\alpha+1)(c_{1}-c_{2})} \biggr], \\ & \mathrm{A}_{2}(\alpha)= \biggl[\frac{-c_{2}^{\alpha+1}}{(\alpha+1)(c_{1}-c_{2})}+ \frac{c_{1}^{\alpha+2}-c_{2}^{\alpha+2}}{(\alpha+1)(\alpha +2)(c_{1}-c_{2})^{2}}-\frac{c_{1}^{\alpha}}{2} \biggr], \\ & \mathrm{B}_{2}(\alpha)= \biggl[\frac{c_{2}^{\alpha}}{2}+\frac {c_{2}^{\alpha+1}}{(\alpha+1)(c_{1}-c_{2})} +\frac{c_{1}^{\alpha+2}-c_{2}^{\alpha+2}}{(\alpha+1)(\alpha +2)(c_{1}-c_{2})^{2}} \biggr], \\ & \mathrm{A}_{3}(\alpha)= \biggl[\frac{c_{1}^{\alpha+1}}{(\alpha+1)(c_{1}-c_{2})} - \frac{c_{1}^{\alpha+2}-c_{2}^{\alpha+2}}{(\alpha+1)(\alpha +2)(c_{1}-c_{2})^{2}}-\frac{c_{1}^{\alpha}}{2} \biggr], \\ & \mathrm{B}_{3}(\alpha)= \biggl[\frac{c_{2}^{\alpha}}{2}-\frac {c_{1}^{\alpha+1}}{(\alpha+1)(c_{1}-c_{2})}+ \frac{c_{1}^{\alpha+2}-c_{2}^{\alpha+2}}{(\alpha+1)(\alpha +2)(c_{1}-c_{2})^{2}} \biggr]. \end{aligned}$$
Proof
From Lemma 2.1 and the well-known Hölder mean inequality together with the convexity of \(\vert \psi^{\prime} \vert ^{q}\) on the interval \([c_{1}, c_{2}]\) we clearly see that
$$\begin{aligned} & \biggl\vert \frac{\psi(c_{1})+\psi(c_{2})}{2}-\frac{\alpha }{c_{2}^{\alpha}-c_{1}^{\alpha}} \int_{c_{1}}^{c_{2}}\psi(\xi)\,d_{\alpha }\xi \biggr\vert \\ &\quad = \biggl\vert \frac{c_{2}-c_{1}}{2(c_{2}^{\alpha}-c_{1}^{\alpha})} \biggl[ \int_{0}^{1} \bigl(\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{2\alpha-1}-c_{1}^{\alpha}\bigl( \theta c_{1}+(1-\theta)c_{2}\bigr)^{\alpha-1} \bigr) \\ &\quad\quad{}\times \mathrm{D}_{\alpha}(\psi) \bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)\,d\theta \\ &\quad\quad{} + \int_{0}^{1} \bigl(\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{2\alpha-1}-c_{2}^{\alpha}\bigl( \theta c_{1}+(1-\theta)c_{2}\bigr)^{\alpha-1} \bigr) \mathrm{D}_{\alpha}(\psi) \bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)\,d\theta \biggr] \biggr\vert \\ &\quad \leq\frac{c_{2}-c_{1}}{2(c_{2}^{\alpha}-c_{1}^{\alpha})} \biggl[ \int _{0}^{1} \bigl(\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{\alpha}-c_{1}^{\alpha} \bigr) \bigl\vert \psi ^{\prime}\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr) \bigr\vert \,d\theta \\ &\quad\quad{} + \int_{0}^{1} \bigl(c_{2}^{\alpha}- \bigl(\theta c_{1}+(1-\theta)c_{2}\bigr)^{\alpha} \bigr) \bigl\vert \psi^{\prime}\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr) \bigr\vert \,d\theta \biggr], \end{aligned}$$
(2.4)
$$\begin{aligned} & \int_{0}^{1} \bigl(\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{\alpha}-c_{1}^{\alpha} \bigr) \bigl\vert \psi ^{\prime}\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr) \bigr\vert \,d\theta \\ &\quad \leq \biggl( \int_{0}^{1} \bigl(\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{\alpha}-c_{1}^{\alpha} \bigr)\,d\theta \biggr)^{1-1/q} \\ &\quad\quad{} \times \biggl( \int_{0}^{1} \bigl(\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{\alpha}-c_{1}^{\alpha} \bigr) \bigl\vert \psi '\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr) \bigr\vert ^{q}\,d\theta \biggr)^{1/q}, \end{aligned}$$
(2.5)
$$\begin{aligned} & \int_{0}^{1} \bigl(c_{2}^{\alpha}- \bigl(\theta c_{1}+(1-\theta)c_{2}\bigr)^{\alpha} \bigr) \bigl\vert \psi^{\prime}\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr) \bigr\vert \,d\theta \\ &\quad \leq \biggl( \int_{0}^{1} \bigl( c_{2}^{\alpha}- \bigl(\theta c_{1}+(1-\theta)c_{2}\bigr)^{\alpha} \bigr)\,d\theta \biggr)^{1-1/q} \\ &\quad\quad{} \times \biggl( \int_{0}^{1} \bigl(c_{2}^{\alpha}- \bigl(\theta c_{1}+(1-\theta)c_{2}\bigr)^{\alpha} \bigr) \bigl\vert \psi^{\prime}\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr) \bigr\vert ^{q}\,d\theta \biggr)^{1/q}, \end{aligned}$$
(2.6)
$$\begin{aligned} & \int_{0}^{1} \bigl(\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{\alpha}-c_{1}^{\alpha} \bigr) \bigl\vert \psi ^{\prime}\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr) \bigr\vert ^{q}\,d\theta \\ &\quad \leq \int_{0}^{1} \bigl(\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{\alpha}-c_{1}^{\alpha} \bigr) \bigl[(1-\theta) \bigl\vert \psi^{\prime}(c_{1}) \bigr\vert ^{q}+\theta \bigl\vert \psi^{\prime }(c_{2}) \bigr\vert ^{q} \bigr]\,d\theta \\ &\quad = \bigl\vert \psi^{\prime}(c_{1}) \bigr\vert ^{q} \int_{0}^{1} \bigl(\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{\alpha}-c_{1}^{\alpha} \bigr) (1-\theta)\,d\theta \\ &\quad\quad{} + \bigl\vert \psi^{\prime}(c_{2}) \bigr\vert ^{q} \int_{0}^{1} \bigl(\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{\alpha}-c_{1}^{\alpha} \bigr) \theta \,d\theta \\ &\quad = \bigl\vert \psi^{\prime}(c_{1}) \bigr\vert ^{q} \biggl[\frac {-c_{2}^{\alpha+1}}{(\alpha+1)(c_{1}-c_{2})}+ \frac{c_{1}^{\alpha+2}-c_{2}^{\alpha+2}}{(\alpha+1)(\alpha +2)(c_{1}-c_{2})^{2}}-\frac{c_{1}^{\alpha}}{2} \biggr] \\ &\quad\quad{} + \bigl\vert \psi^{\prime}(c_{2}) \bigr\vert ^{q} \biggl[\frac{\alpha +1}{(\alpha+1)(c_{1}-c_{2})} -\frac{c_{1}^{\alpha+2}-c_{2}^{\alpha+2}}{(\alpha+1)(\alpha +2)(c_{1}-c_{2})^{2}}-\frac{c_{1}^{\alpha}}{2} \biggr], \end{aligned}$$
(2.7)
$$\begin{aligned} & \int_{0}^{1} \bigl(c_{2}^{\alpha}- \bigl(\theta c_{1}+(1-\theta)c_{2}\bigr)^{\alpha} \bigr) \bigl\vert \psi^{\prime}\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr) \bigr\vert ^{q}\,d\theta \\ &\quad \leq \int_{0}^{1} \bigl(c_{2}^{\alpha}- \bigl(\theta c_{1}+(1-\theta)c_{2}\bigr)^{\alpha} \bigr) \bigl[(1-\theta) \bigl\vert \psi ^{\prime}(c_{1}) \bigr\vert ^{q}+\theta \bigl\vert \psi^{\prime }(c_{2}) \bigr\vert ^{q} \bigr]\,d\theta \\ &\quad = \bigl\vert \psi^{\prime}(c_{1}) \bigr\vert ^{q} \int_{0}^{1} \bigl(c_{2}^{\alpha}- \bigl(\theta c_{1}+(1-\theta)c_{2}\bigr)^{\alpha} \bigr) (1-\theta)\,d\theta \\ &\quad\quad{} + \bigl\vert \psi^{\prime}(c_{2}) \bigr\vert ^{q} \int_{0}^{1} \bigl(c_{2}^{\alpha}- \bigl(\theta c_{1}+(1-\theta)c_{2}\bigr)^{\alpha} \bigr)\theta \,d\theta \\ &\quad = \bigl\vert \psi^{\prime}(c_{1}) \bigr\vert ^{q} \biggl[\frac {c_{2}^{\alpha}}{2}+\frac{c_{2}^{\alpha+1}}{(\alpha+1) (c_{1}-c_{2})}+\frac{c_{1}^{\alpha+2}-c_{2}^{\alpha+2}}{(\alpha +1)(\alpha+2)(c_{1}-c_{2})^{2}} \biggr] \\ &\quad\quad{} + \bigl\vert \psi^{\prime}(c_{2}) \bigr\vert ^{q} \biggl[\frac {c_{2}^{\alpha}}{2}-\frac{c_{1}^{\alpha+1}}{ (\alpha+1)(c_{1}-c_{2})}+\frac{c_{1}^{\alpha+2}-c_{2}^{\alpha +2}}{(\alpha+1)(\alpha+2)(c_{1}-c_{2})^{2}} \biggr]. \end{aligned}$$
(2.8)
Therefore, inequality (2.3) follows easily from (2.4)-(2.8). □
Remark 2.6
Let \(\alpha=1\). Then inequality (2.3) becomes
$$\begin{aligned} & \biggl\vert \frac{\psi(c_{1})+\psi(c_{2})}{2}-\frac{1}{c_{2}-c_{1}} \int _{c_{1}}^{c_{2}}\psi(\xi)\,d\xi \biggr\vert \\ &\quad \leq\frac{1}{2} \biggl(\frac{c_{2}-c_{1}}{2} \biggr)^{1-1/q} \bigl[ \bigl\{ \mathrm{A}_{2}(1) \bigl\vert \psi^{\prime}(c_{1}) \bigr\vert ^{q}+ \mathrm{A}_{3}(1) \bigl\vert \psi^{\prime}(c_{2}) \bigr\vert ^{q} \bigr\} ^{1/q} \\ &\quad\quad{} + \bigl\{ \mathrm{B}_{2}(1) \bigl\vert \psi^{\prime}(c_{1}) \bigr\vert ^{q}+\mathrm{B}_{3}(1) \bigl\vert \psi^{\prime}(c_{2}) \bigr\vert ^{q} \bigr\} ^{1/q} \bigr] \end{aligned}$$
with
$$\begin{aligned}& \mathrm{A}_{2}(1)=\frac{c_{2}-c_{1}}{3},\quad\quad \mathrm{B}_{2}(1)= \frac {(c_{1}+c_{2})^{2}+2c_{1}c_{2}}{6(c_{1}-c_{2})}, \\ & \mathrm{A}_{3}(1)=\frac{c_{2}-c_{1}}{6}, \quad \quad \mathrm{B}_{3}(1)= \frac{c_{2}-c_{1}}{3}. \end{aligned}$$
Theorem 2.7
Let
\(\alpha\in(0, 1]\), \(q>1\), \(c_{1}, c_{2}\in\mathbb{R} \)
with
\(0\leq c_{1} < c_{2}\)
and
\(\psi:[c_{1}, c_{2}] \rightarrow \mathbb{R}\)
be an
α-differentiable function on
\((c_{1}, c_{2})\). Then the inequality
$$\begin{aligned} & \biggl\vert \frac{\psi(c_{1})+\psi(c_{2})}{2}-\frac{\alpha }{c_{2}^{\alpha}-c_{1}^{\alpha}} \int_{c_{1}}^{c_{2}}\psi(\xi)\,d_{\alpha }\xi \biggr\vert \\ &\quad \leq\frac{c_{2}-c_{1}}{2(c_{2}^{\alpha}-c_{1}^{\alpha})} \biggl[ \mathrm{A}_{1}(\alpha) \psi^{\prime} \biggl(\frac{\mathrm{C}_{1}(\alpha )}{\mathrm{A}_{1}(\alpha)} \biggr) +\mathrm{B}_{1}( \alpha)\psi^{\prime} \biggl(\frac{\mathrm{C}_{2}(\alpha )}{\mathrm{B}_{1}(\alpha)} \biggr) \biggr] \end{aligned}$$
(2.9)
holds if
\(\mathrm{D}_{\alpha}(\psi)\in \mathrm{L}_{\alpha}^{1}([c_{1}, c_{2}])\)
and
\(\vert \psi^{\prime } \vert ^{q}\)
is concave on
\([c_{1}, c_{2}]\), where
\(\mathrm{A}_{1}(\alpha)\)
and
\(\mathrm{B}_{1}(\alpha)\)
are defined as in Theorem
2.5, and
\(\mathrm{C}_{1}(\alpha)\)
and
\(\mathrm{C}_{2}(\alpha)\)
are defined by
$$ \mathrm{C}_{1}(\alpha)= \biggl[\frac{c_{1}^{\alpha+2}-c_{2}^{\alpha +2}}{(\alpha+2)(c_{1}-c_{2})}-\frac{c_{1}^{\alpha}(c_{1}-c_{2})}{2} \biggr],\quad \quad \mathrm{C}_{2}(\alpha)= \biggl[\frac{c_{2}^{\alpha}(c_{1}+c_{2})}{2}- \frac {c_{1}^{\alpha+2}-c_{2}^{\alpha+2}}{(\alpha+2)(c_{1}-c_{2})} \biggr]. $$
Proof
It follows from the concavity of \(\vert \psi' \vert ^{q}\) and the Hölder mean inequality that
$$\begin{aligned}& \bigl(\theta \bigl\vert \psi^{\prime}(c_{1}) \bigr\vert +(1- \theta) \bigl\vert \psi^{\prime}(c_{2}) \bigr\vert \bigr)^{q}\leq\theta \bigl\vert \psi ^{\prime}(c_{1}) \bigr\vert ^{q}+(1-\theta) \bigl\vert \psi^{\prime }(c_{2}) \bigr\vert ^{q} \leq \bigl\vert \psi^{\prime}\bigl(\theta c_{1}+(1-\theta)c_{2}\bigr) \bigr\vert ^{q}, \\& \bigl\vert \psi^{\prime}\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr) \bigr\vert \geq \theta \bigl\vert \psi^{\prime}(c_{1}) \bigr\vert +(1-\theta) \bigl\vert \psi^{\prime}(c_{2}) \bigr\vert , \end{aligned}$$
which implies that \(\vert \psi' \vert \) is also concave. Making use of Lemma 2.1 and the Jensen integral inequality, we have
$$\begin{aligned} & \biggl\vert \frac{\psi(c_{1})+\psi(c_{2})}{2}-\frac{\alpha }{c_{2}^{\alpha}-c_{1}^{\alpha}} \int_{c_{1}}^{c_{2}}\psi(\xi)\,d_{\alpha }\xi \biggr\vert \\ &\quad = \biggl\vert \frac{c_{2}-c_{1}}{2(c_{2}^{\alpha}-c_{1}^{\alpha})} \biggl[ \int_{0}^{1} \bigl(\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{2\alpha-1}-c_{1}^{\alpha}\bigl( \theta c_{1}+(1-\theta)c_{2}\bigr)^{\alpha-1} \bigr) \\ &\quad\quad{}\times \mathrm{D} _{\alpha}(\psi) \bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)\,d\theta \\ &\quad\quad{} + \int_{0}^{1} \bigl(\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{2\alpha-1}-c_{2}^{\alpha}\bigl( \theta c_{1}+(1-\theta)c_{2}\bigr)^{\alpha-1} \bigr) \mathrm{D}_{\alpha}(\psi) \bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)\,d\theta \biggr] \biggr\vert \\ &\quad \leq\frac{c_{2}-c_{1}}{2(c_{2}^{\alpha}-c_{1}^{\alpha})} \biggl[ \int _{0}^{1} \bigl(\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{\alpha}-c_{1}^{\alpha} \bigr) \bigl\vert \psi ^{\prime}\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr) \bigr\vert \,d\theta \\ &\quad\quad{} + \int_{0}^{1} \bigl(c_{2}^{\alpha}- \bigl(\theta c_{1}+(1-\theta)c_{2}\bigr)^{\alpha} \bigr) \bigl\vert \psi^{\prime}\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr) \bigr\vert \,d\theta \biggr], \end{aligned}$$
(2.10)
$$\begin{aligned} & \int_{0}^{1} \bigl(\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{\alpha}-c_{1}^{\alpha} \bigr) \bigl\vert \psi ^{\prime}\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr) \bigr\vert \,d\theta \\ &\quad \leq \biggl( \int_{0}^{1} \bigl(\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{\alpha}-c_{1}^{\alpha} \bigr)\,d\theta \biggr) \\ &\quad\quad{} \times\psi^{\prime} \biggl(\frac{\int_{0}^{1} ((\theta c_{1}+(1-\theta)c_{2})^{\alpha}-c_{1}^{\alpha} ) (\theta c_{1}+(1-\theta)c_{2})\,d\theta}{\int_{0}^{1} ((\theta c_{1}+(1-\theta)c_{2})^{\alpha}-c_{1}^{\alpha} )\,d\theta} \biggr) \\ &\quad =\mathrm{A}_{1}(\alpha)\psi^{\prime} \biggl(\frac{\mathrm{C}_{1}(\alpha )}{\mathrm{A}_{1}(\alpha)} \biggr), \end{aligned}$$
(2.11)
$$\begin{aligned} & \int_{0}^{1} \bigl(c_{2}^{\alpha}- \bigl(\theta c_{1}+(1-\theta)c_{2}\bigr)^{\alpha} \bigr) \bigl\vert \psi^{\prime}\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr) \bigr\vert \,d\theta \\ &\quad \leq \biggl(c_{2}^{\alpha}- \int_{0}^{1} \bigl(\bigl(\theta c_{1}+(1- \theta)c_{2}\bigr)^{\alpha} \bigr)\,d\theta \biggr) \\ &\quad\quad{} \times\psi^{\prime} \biggl(\frac{\int_{0}^{1} (c_{2}^{\alpha}-(\theta c_{1}+(1-\theta)c_{2})^{\alpha} )(\theta c_{1}+(1-\theta)c_{2})\,d\theta}{ \int_{0}^{1} (c_{2}^{\alpha}-(\theta c_{1}+(1-\theta)c_{2})^{\alpha} )\,d\theta} \biggr) \\ &\quad =\mathrm{B}_{1}(\alpha)\psi^{\prime} \biggl(\frac{\mathrm{C}_{2}(\alpha )}{\mathrm{B}_{1}(\alpha)} \biggr). \end{aligned}$$
(2.12)
Therefore, inequality (2.9) follows easily from (2.10)-(2.12). □
Remark 2.8
Let \(\alpha=1\). Then inequality (2.9) leads to
$$\begin{aligned} & \biggl\vert \frac{\psi(c_{1})+\psi(c_{2})}{2}-\frac{1}{c_{2}-c_{1}} \int _{c_{1}}^{c_{2}}\psi(\xi)\,d\xi \biggr\vert \\ &\quad \leq\frac{c_{2}-c_{1}}{4} \biggl[\psi^{\prime} \biggl(\frac {2c_{2}^{2}-c_{1}^{2}+5c_{1}c_{2}}{3(c_{2}-c_{1})} \biggr) +\psi^{\prime} \biggl(\frac{c_{2}^{2}-2c_{1}^{2}+c_{1} c_{2}}{3(c_{2}-c_{1})} \biggr) \biggr]. \end{aligned}$$