Skip to content

Advertisement

Journal of Inequalities and Applications
Journal of Inequalities and Applications Cover Image
  • Research
  • Open Access

Inequalities for α-fractional differentiable functions

  • 1Email author,
  • 2,
  • 2 and
  • 3
Journal of Inequalities and Applications20172017:93

https://doi.org/10.1186/s13660-017-1371-6

©  The Author(s) 2017

  • Received: 8 October 2016
  • Accepted: 20 April 2017
  • Published:

Abstract

In this article, we present an identity and several Hermite-Hadamard type inequalities for conformable fractional integrals. As applications, we establish some inequalities for certain special means of two positive real numbers and give the error estimations for the trapezoidal formula.

Keywords

  • convex function
  • Hermite-Hadamard inequality
  • fractional derivative
  • fractional integral
  • special mean
  • trapezoidal formula

MSC

  • 26D15
  • 26A51
  • 26A33

1 Introduction

A real-valued function \(\psi: \mathrm{I}\subseteq \mathbb{R}\rightarrow\mathbb{R}\) is said to be convex on I if the inequality
$$ \psi\bigl(\theta\xi+(1-\theta)\zeta\bigr)\leq\theta \psi(\xi)+(1-\theta)\psi( \zeta) $$
(1.1)
holds for all \(\xi, \zeta\in\mathrm{I}\) and \(\theta\in[0, 1]\). ψ is said to be concave on I if inequality (1.1) is reversed.
Let \(\psi: \mathrm{I}\subseteq\mathbb{R} \rightarrow\mathbb{R}\) be a convex function on the interval I, and \(c_{1}, c_{2} \in\mathrm{I}\) with \(c_{1}< c_{2}\). Then the double inequality
$$ \psi \biggl(\frac{c_{1}+c_{2}}{2} \biggr)\leq\frac{1}{c_{2}-c_{1}} \int _{c_{1}}^{c_{2}}\psi(\xi)\,d\xi\leq\frac{\psi(c_{1})+\psi(c_{2})}{2} $$
(1.2)
is known in the literature as the Hermite-Hadamard inequality for convex functions [1–3]. Both inequalities hold in the reversed direction if ψ is concave on the interval I. In particular, many classical inequalities for means can be derived from (1.2) for appropriate particular selections of the function ψ.

Recently, the improvements, generalizations, refinements and applications for the Hermite-Hadamard inequality have attracted the attention of many researchers [4–22].

Dragomir and Agarwal [23] proved the following results connected with the right hand part of (1.2).

Theorem 1.1

See [23], Lemma 2.1

Let \(\psi:\mathrm{I}^{\circ}\subseteq\mathbb{R}\rightarrow \mathbb{R}\) be a differentiable mapping on  \(\mathrm{I}^{\circ}\). Then the identity
$$ \frac{\psi(c_{1})+\psi(c_{2})}{2}-\frac{1}{c_{2}-c_{1}} \int _{c_{1}}^{c_{2}}\psi(\xi)\,d\xi =\frac{c_{2}-c_{1}}{2} \int_{0}^{1}(1-2\theta)\psi^{\prime}\bigl( \theta c_{1}+(1-\theta)c_{2}\bigr)\,d\theta $$
holds for all \(c_{1}, c_{2}\in\mathrm{I}^{\circ}\) with \(c_{1}< c_{2}\) if \(\psi^{\prime}\in\mathrm{L}[c_{1}, c_{2}]\), where \(\mathrm{I}^{\circ}\) denotes the interior of I.

Theorem 1.2

See [23], Theorem 2.2

Let \(\psi:\mathrm{I}^{\circ}\subseteq\mathbb{R}\rightarrow \mathbb{R}\) be a differentiable mapping on  \(\mathrm{I}^{\circ}\). Then the inequality
$$ \biggl\vert \frac{\psi(c_{1})+\psi(c_{2})}{2}-\frac{1}{c_{2}-c_{1}} \int _{c_{1}}^{c_{2}} \psi(\xi)\,d\xi \biggr\vert \leq \frac{(c_{2}-c_{1})( \vert \psi^{\prime }(c_{1}) \vert + \vert \psi^{\prime}(c_{2}) \vert )}{8} $$
holds for \(c_{1}, c_{2}\in\mathrm{I}^{\circ}\) with \(c_{1}< c_{2}\) if \(\vert \psi^{\prime} \vert \) is convex on \([c_{1}, c_{2}]\).

Making use of Theorem 1.1, Pearce and Pečarić [24] established Theorem 1.3 as follows.

Theorem 1.3

See [24], Theorem 1

Let \(c_{1}, c_{2}\in\mathrm{I}\subseteq\mathbb{R}\) with \(c_{1}< c_{2}\), \(\psi:\mathrm{I}^{\circ}\subseteq \mathbb{R}\rightarrow\mathbb{R}\) be a differentiable mapping on \(\mathrm{I}^{\circ}\) and \(q\geq1\). Then the inequality
$$ \biggl\vert \frac{\psi(c_{1})+\psi(c_{2})}{2}-\frac{1}{c_{2}-c_{1}} \int_{c_{1}}^{c_{2}}\psi(\xi)\,d\xi \biggr\vert \leq \frac{c_{2}-c_{1}}{4} \biggl[\frac{ \vert \psi^{\prime}(c_{1}) \vert ^{q}+ \vert \psi^{\prime}(c_{2}) \vert ^{q}}{2} \biggr]^{1/q} $$
is valid if the mapping \(\vert \psi^{\prime} \vert ^{q}\) is convex on the interval \([c_{1}, c_{2}]\).

Next, we recall several elementary definitions and important results in the theory of conformable fractional calculus, which will be used throughout the article, we refer the interested reader to [25–32].

The conformable fractional derivative of order \(0<\alpha\leq1\) for a function \(\psi: (0, \infty)\rightarrow\mathbb{R}\) at \(\xi>0\) is defined by
$$ \mathrm{D}_{\alpha}(\psi) (\xi)=\lim_{\epsilon\rightarrow 0} \frac{\psi(\xi+\epsilon\xi^{1-\alpha})-\psi(\xi)}{\epsilon}, $$
and the fractional derivative at 0 is defined as \(\mathrm{D}_{\alpha}(\psi)(0)=\lim_{\xi\rightarrow 0^{+}}\mathrm{D}_{\alpha}(\psi)(\xi)\).
The (left) fractional derivative starting from \(c_{1}\) of a function \(\psi: [c_{1}, \infty)\rightarrow\mathbb{R}\) of order \(0<\alpha \leq1\) is defined by
$$ \mathrm{D}_{\alpha}^{c_{1}}(\psi) (\xi)=\lim_{\epsilon\rightarrow 0} \frac{\psi(\xi+\epsilon (\xi-c_{1})^{1-\alpha})-\psi(\xi)}{\epsilon}, $$
and we write \(\mathrm{D}_{\alpha}^{c_{1}}(\psi)=\mathrm{D}_{\alpha}^{0}(\psi)=\mathrm {D}_{\alpha}(\psi)\) if \(c_{1}=0\). For more details see [26].
Let \(\alpha\in(0, 1]\) and \(\psi, \phi\) be α-differentiable at \(\xi>0\). Then we have
$$\begin{aligned}& \frac{d_{\alpha}}{d_{\alpha}\xi} \bigl(\xi^{n} \bigr) =n\xi^{n-\alpha}, \quad n \in\mathbb{R}, \\& \frac{d_{\alpha}}{d_{\alpha}\xi}(c) =0, \quad c\in\mathbb{R}, \\& \frac{d_{\alpha}}{d_{\alpha}\xi} \bigl(c_{1} \psi(\xi)+ c_{2} \phi(\xi) \bigr) =c_{1} \frac{d_{\alpha}}{d_{\alpha}\xi}\bigl(\psi(\xi)\bigr)+c_{2} \frac{d_{\alpha}}{d_{\alpha}\xi}\bigl(\phi(\xi)\bigr), \quad c_{1}, c_{2} \in\mathbb{R}, \\& \frac{d_{\alpha}}{d_{\alpha}\xi} \bigl(\psi(\xi)\phi(\xi) \bigr) =\psi(\xi) \frac{d_{\alpha}}{d_{\alpha}\xi} \bigl(\phi(\xi)\bigr)+\phi(\xi) \frac{d_{\alpha}}{d_{\alpha}\xi}\bigl(\psi(\xi)\bigr), \\& \frac{d_{\alpha}}{d_{\alpha}\xi} \biggl(\frac{\psi(\xi)}{\phi(\xi)} \biggr) =\frac{\phi(\xi) \frac{d_{\alpha}}{d_{\alpha}\xi}(\psi(\xi))-\phi(\xi) \frac{d_{\alpha}}{d_{\alpha}\xi}(\psi(\xi))}{(\phi(\xi))^{2}}, \\& \frac{d_{\alpha}}{d_{\alpha}\xi} \bigl(\psi(\phi) (\xi) \bigr)=\psi'\bigl(\phi ( \xi)\bigr)\frac{d_{\alpha}}{d_{\alpha}\xi}\bigl(\phi(\xi)\bigr), \end{aligned}$$
(1.3)
where ψ is differentiable at \(\phi(\xi)\) in equation (1.3). In particular,
$$ \frac{d_{\alpha}}{d_{\alpha}\xi} \bigl(\psi(\xi) \bigr)=\xi^{1-\alpha} \frac{d}{d\xi} \bigl(\psi(\xi)\bigr) $$
if ψ is differentiable.
Let \(\alpha\in(0, 1] \) and \(0\leq c_{1} < c_{2}\). A function \(\psi : [c_{1}, c_{2}] \rightarrow\mathbb{R} \) is said to be α-fractional integrable on \([c_{1}, c_{2}]\) if the integral
$$ \int_{c_{1}}^{c_{2}}\psi(\xi)\,d_{\alpha}\xi= \int_{c_{1}}^{c_{2}}\psi(\xi )\xi^{\alpha-1}\,d\xi $$
exists and is finite. All the α-fractional integrable functions on \([c_{1}, c_{2}]\) are denoted by \(\mathrm{L}_{\alpha}^{1}([c_{1}, c_{2}])\).
It is well known that
$$ \int_{c_{1}}^{c_{2}}\psi(\xi)\mathrm{D}_{\alpha}^{c_{1}}( \phi) (\xi )\,d_{\alpha}\xi=\psi\phi| _{c_{1}}^{c_{2}} - \int_{c_{1}}^{c_{2}}\phi(\xi)\mathrm{D}_{\alpha}^{c_{1}}( \psi) (\xi )\,d_{\alpha}\xi $$
if \(\psi, \phi: [c_{1}, c_{2}] \rightarrow\mathbb{R}\) are two functions such that ψϕ is differentiable.

Very recently, Anderson [33] established a Hermite-Hadamard type inequality for fractional differentiable functions as follows.

Theorem 1.4

Let \(\alpha\in(0, 1]\) and \(\psi: [c_{1}, c_{2}]\rightarrow\mathbb{R}\) be an α-fractional differentiable function. Then the inequality
$$ \frac{\alpha}{c_{2}^{\alpha}-c_{1}^{\alpha}} \int_{c_{1}}^{c_{2}}\psi(\xi )\,d_{\alpha}\xi\leq \frac{\psi(c_{1})+\psi(c_{2})}{2} $$
(1.4)
holds if \(\mathrm{D}_{\alpha}(\psi)\) is increasing on \([c_{1}, c_{2}]\). Moreover, if the function ψ is decreasing on \([c_{1}, c_{2}]\), then one has
$$ \psi \biggl(\frac{c_{1}+c_{2}}{2} \biggr)\leq\frac{\alpha}{c_{2}^{\alpha }-c_{1}^{\alpha}} \int_{c_{1}}^{c_{2}}\psi(\xi)\,d_{\alpha}\xi. $$
(1.5)

Remark 1.5

We clearly see that inequalities (1.4) and (1.5) reduce to inequality (1.2) if \(\alpha=1\).

The main purpose of the article is to present an identity and several Hermite-Hadamard type inequalities for conformable fractional integrals, establish some inequalities for certain special means of two positive real numbers and give the error estimations for the trapezoidal formula.

2 Main results

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
$$\begin{aligned} & \biggl\vert \frac{\psi(c_{1})+\psi(c_{2})}{2}-\frac{1}{c_{2}-c_{1}} \int _{c_{1}}^{c_{2}}\psi(\xi)\,d\xi \biggr\vert \\ &\quad \leq\frac{c_{2}-c_{1}}{4} \biggl[\psi^{\prime} \biggl(\frac {2c_{2}^{2}-c_{1}^{2}+5c_{1}c_{2}}{3(c_{2}-c_{1})} \biggr) +\psi^{\prime} \biggl(\frac{c_{2}^{2}-2c_{1}^{2}+c_{1} c_{2}}{3(c_{2}-c_{1})} \biggr) \biggr]. \end{aligned}$$

3 Applications to special means of real numbers

Let \(\alpha\in(0,1]\), \(r\in\mathbb{R}\), \(r\neq0, -\alpha\) and \(a, b>0\) with \(a\neq b\). Then the arithmetic mean \(\mathrm{A}(a, b)\), logarithmic mean \(\mathrm{L}(a, b)\) and \((\alpha, r)\)th generalized logarithmic mean \(\mathrm{L}_{(\alpha, r)}(a,b)\) of a and b are defined by
$$ \mathrm{A}(a,b)=\frac{a+b}{2}, \quad \quad\mathrm{L}(a,b)=\frac{a-b}{\log a-\log b}, \quad\quad \mathrm{L}_{(\alpha, r)}(a,b)= \biggl[\frac{\alpha (b^{\alpha+r}-a^{\alpha+r} )}{(\alpha+r) (b^{\alpha}-a^{\alpha} )} \biggr]^{1/r}, $$
respectively. Then from Theorems 2.3 and 2.5 together with the convexities of the functions \(\xi\rightarrow\xi^{r}\) and \(\xi\rightarrow1/\xi\) on the interval \((0, \infty)\) we get several new inequalities for the arithmetic, logarithmic and generalized logarithmic means as follows.

Theorem 3.1

Let \(c_{1},c_{2}\in\mathbb{R}\) with \(0< c_{1}< c_{2}\), \(r>1\), \(q>1\) and \(\alpha\in(0, 1]\). Then we have
$$\begin{aligned}& \bigl\vert \mathrm{A}\bigl(c_{1}^{r},c_{2}^{r} \bigr)-\mathrm{L}^{r}_{(\alpha ,r)}(c_{1},c_{2}) \bigr\vert \\& \quad \leq\frac{r(c_{2}-c_{1}) (5c_{2}^{\alpha}-7c_{1}^{\alpha }+c_{1}c_{2}^{\alpha-1}+c_{1}^{\alpha-1}c_{2} )}{ 12(c_{2}^{\alpha}-c_{1}^{\alpha})}\mathrm{A}\bigl( \vert c_{1} \vert ^{r-1}, \vert c_{2} \vert ^{r-1}\bigr), \\& \bigl\vert \mathrm{A}\bigl(c_{1}^{r},c_{2}^{r} \bigr)-\mathrm{L}^{r}_{(\alpha ,r)}(c_{1},c_{2})) \bigr\vert \\& \quad \leq \frac{r(c_{2}-c_{1})}{2(c_{2}^{\alpha}-c_{1}^{\alpha})} \bigl[ \bigl(\mathrm{A}_{1} (\alpha) \bigr)^{1-1/q} \bigl\{ \mathrm{A}_{2}(\alpha) \vert c_{1} \vert ^{(r-1)q} +\mathrm{A}_{3}(\alpha) \vert c_{2} \vert ^{(r-1)q} \bigr\} ^{1/q} \\& \quad\quad{} + \bigl(\mathrm{B}_{1}(\alpha) \bigr)^{1-1/q} \bigl\{ \mathrm{B}_{2} (\alpha) \vert c_{1} \vert ^{(r-1)q}+\mathrm{B}_{3}(\alpha ) \vert c_{2} \vert ^{(r-1)q} \bigr\} ^{1/q} \bigr], \\& \bigl\vert \mathrm{A}\bigl(c_{1}^{-1},c_{2}^{-1} \bigr)-\mathrm{L}^{-1}_{(\alpha ,-1)}(c_{1},c_{2}) \bigr\vert \leq \frac{(c_{2}-c_{1}) (5c_{2}^{\alpha}-7c_{1}^{\alpha }+c_{1}c_{2}^{\alpha-1}+c_{1}^{\alpha-1}c_{2} )}{ 12(c_{2}^{\alpha}-c_{1}^{\alpha})}\mathrm{A}\bigl(c_{1}^{-2}, c_{2}^{-2}\bigr), \\& \bigl\vert \mathrm{A}\bigl(c_{1}^{-1},c_{2}^{-1} \bigr)-\mathrm{L}^{-1}_{(\alpha, -1)}(c_{1},c_{2}) \bigr\vert \leq \frac{c_{2}-c_{1}}{2(c_{2}^{\alpha}-c_{1}^{\alpha})} \bigl[ \bigl(\mathrm {A}_{1}( \alpha) \bigr)^{1-1/q} \bigl\{ \mathrm{A}_{2}(\alpha) \vert c_{1} \vert ^{-2q}+\mathrm {A}_{3}(\alpha) \vert c_{2} \vert ^{-2q} \bigr\} ^{1/q} \\& \hphantom{\bigl\vert \mathrm{A}\bigl(c_{1}^{-1},c_{2}^{-1} \bigr)-\mathrm{L}^{-1}_{(\alpha, -1)}(c_{1},c_{2}) \bigr\vert \leq}{}+ \bigl(\mathrm{B}_{1}(\alpha) \bigr)^{1-1/q} \bigl\{ \mathrm {B}_{2}(\alpha) \vert c_{1} \vert ^{-2q} + \mathrm{B}_{3}(\alpha) \vert c_{2} \vert ^{-2q} \bigr\} ^{\frac {1}{q}} \bigr], \end{aligned}$$
where \(\mathrm{A}_{1}(\alpha)\), \(\mathrm{A}_{2}(\alpha)\), \(\mathrm{A}_{3}(\alpha)\), \(\mathrm{B}_{1}(\alpha)\), \(\mathrm{B}_{2}(\alpha)\) and \(\mathrm{B}_{3}(\alpha)\) are defined as in Theorem 2.5.

4 Applications to the trapezoidal formula

Let Δ be a division \(c_{1}=\xi_{0}<\xi_{1}<\cdots<\xi_{n-1}<\xi_{n}=c_{2}\) of the interval \([c_{1}, c_{2}]\) and consider the quadrature formula
$$ \int_{c_{1}}^{c_{2}}\psi(\xi)\,d_{\alpha}\xi= \mathrm{T}_{\alpha}(\psi, \Delta)+\mathrm{E}_{\alpha}(\psi, \Delta), $$
where
$$ \mathrm{T}_{\alpha}(\psi, \Delta)=\sum_{i=0}^{n-1} \frac{\psi(\xi_{i})+\psi(\xi_{i+1})}{ 2}\frac{ (\xi^{\alpha}_{i+1}-\xi^{\alpha}_{i} )}{\alpha} $$
is the trapezoidal version and \(\mathrm{E}_{\alpha}(\psi, \Delta)\) denotes the associated approximation error. In this section, we are going to derive several new error estimations for the trapezoidal formula.

Theorem 4.1

Let \(\alpha\in(0, 1]\), \(c_{1}, c_{2}\in\mathbb{R}\) with \(0\leq c_{1} < c_{2}\), \(\psi:[c_{1}, c_{2}]\rightarrow\mathbb{R}\) be an α-differentiable function on \((c_{1}, c_{2})\) and Δ be a division \(c_{1}=\xi_{0}<\xi_{1}<\cdots<\xi_{n-1}<\xi_{n}=c_{2}\) of the interval \([c_{1}, c_{2}]\). Then the inequality
$$ \bigl\vert \mathrm{E}_{\alpha}(\psi, \Delta) \bigr\vert \leq \frac{1}{12\alpha}\max\bigl\{ \bigl\vert \psi^{\prime}(c_{1}) \bigr\vert , \bigl\vert \psi^{\prime}(c_{2}) \bigr\vert \bigr\} \sum_{i=0}^{n-1}(\xi _{i+1}- \xi_{i}) \bigl(5\xi_{i+1}^{\alpha}-7 \xi_{i}^{\alpha}+\xi_{i}\xi_{i+1}^{\alpha -1}+ \xi_{i}^{\alpha-1}\xi_{i+1} \bigr) $$
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

Applying Theorem 2.3 on the subinterval \([\xi_{i},\xi _{i+1}]\) \((i= 0, 1,\ldots, n - 1)\) of the division Δ, we have
$$\begin{aligned} & \biggl\vert \frac{\psi(\xi_{i})+\psi(\xi_{i+1})}{2}\frac{(\xi^{\alpha }_{i+1}-\xi^{\alpha}_{i})}{ \alpha}- \int_{\xi_{i}}^{\xi_{i+1}}\psi(\xi)\,d_{\alpha}\xi \biggr\vert \\ &\quad \leq\frac{(\xi_{i+1}-\xi_{i})}{2\alpha} \biggl[\frac{( \vert \psi^{\prime}(\xi_{i}) \vert + \vert \psi^{\prime}(\xi_{i+1}) \vert ) (5\xi_{i+1}^{\alpha}-7\xi _{i}^{\alpha}+ \xi_{i}\xi_{i+1}^{\alpha-1}+\xi_{i}^{\alpha-1}\xi_{i+1} )}{12} \biggr]. \end{aligned}$$
(4.1)
It follows from (4.1) and the convexity of \(\vert \psi^{\prime}(\xi ) \vert \) on the interval \([c_{1}, c_{2}]\) that
$$\begin{aligned}& \bigl\vert \mathrm{E}_{\alpha}(\psi, \Delta) \bigr\vert = \biggl\vert \mathrm{T}_{\alpha}(\psi, \Delta)- \int_{c_{1}}^{c_{2}}\psi(\xi)\,d_{\alpha}\xi \biggr\vert \\& \quad = \Biggl\vert \sum_{i=0}^{n-1} \biggl[ \frac{\psi(\xi_{i})+\psi(\xi_{i+1})}{2} \frac{ (\xi^{\alpha}_{i+1}-\xi^{\alpha}_{i} )}{\alpha}- \int _{\xi_{i}}^{\xi_{i+1}}\psi(\xi)\,d_{\alpha}\xi \biggr] \Biggr\vert \\& \quad \leq \sum_{i=0}^{n-1} \biggl\vert \frac{\psi(\xi_{i})+\psi(\xi_{i+1})}{2} \frac{ (\xi^{\alpha}_{i+1}-\xi^{\alpha}_{i} )}{\alpha}- \int _{\xi_{i}}^{\xi_{i+1}}\psi(\xi)\,d_{\alpha}\xi \biggr\vert \\& \quad \leq \frac{1}{2\alpha}\sum_{i=0}^{n-1}( \xi_{i+1}-\xi_{i}) \biggl[\frac{( \vert \psi^{\prime}(\xi_{i}) \vert + \vert \psi^{\prime}(\xi_{i+1}) \vert ) (5\xi_{i+1}^{\alpha}-7\xi _{i}^{\alpha} +\xi_{i}\xi_{i+1}^{\alpha-1}+\xi_{i}^{\alpha-1}\xi_{i+1} )}{12} \biggr] \\& \quad = \frac{1}{12\alpha}\sum_{i=0}^{n-1}( \xi_{i+1}-\xi_{i}) \biggl[\frac{( \vert \psi^{\prime}(\xi_{i}) \vert + \vert \psi^{\prime}(\xi_{i+1}) \vert ) (5\xi_{i+1}^{\alpha}-7\xi _{i}^{\alpha} +\xi_{i}\xi_{i+1}^{\alpha-1}+\xi_{i}^{\alpha-1}\xi_{i+1} )}{2} \biggr] \\& \quad \leq \frac{1}{12\alpha}\sum_{i=0}^{n-1}( \xi_{i+1}-\xi_{i}) \bigl(5\xi _{i+1}^{\alpha}-7 \xi_{i}^{\alpha} +\xi_{i}\xi_{i+1}^{\alpha-1}+ \xi_{i}^{\alpha-1}\xi_{i+1} \bigr)\max\bigl\{ \bigl\vert \psi^{\prime}(\xi_{i}) \bigr\vert , \bigl\vert \psi^{\prime}(\xi_{i+1}) \bigr\vert \bigr\} \\& \quad \leq \frac{1}{12\alpha}\max\bigl\{ \bigl\vert \psi^{\prime}(c_{1}) \bigr\vert , \bigl\vert \psi^{\prime}(c_{2}) \bigr\vert \bigr\} \sum_{i=0}^{n-1}(\xi _{i+1}- \xi_{i}) \bigl(5\xi_{i+1}^{\alpha}-7 \xi_{i}^{\alpha}+\xi_{i}\xi_{i+1}^{\alpha -1}+ \xi_{i}^{\alpha-1}\xi_{i+1} \bigr). \end{aligned}$$
 □

Making use of arguments analogous to the proof of Theorem 4.1, we get Theorem 4.2 immediately.

Theorem 4.2

Let \(\alpha\in(0, 1]\), \(q>1\), \(c_{1}, c_{2}\in\mathbb{R}\) with \(0\leq c_{1} < c_{2}\), \(\psi:[c_{1}, c_{2}]\rightarrow\mathbb{R}\) be an α-differentiable function on \((c_{1}, c_{2})\) and Δ be a division \(c_{1}=\xi_{0}<\xi_{1}<\cdots<\xi_{n-1}<\xi_{n}=c_{2}\) of the interval \([c_{1}, c_{2}]\). Then the inequality
$$\begin{aligned} \bigl\vert \mathrm{E}_{\alpha}(\psi, \Delta) \bigr\vert &\leq\sum _{i=0}^{n-1}\frac{(\xi_{i+1}-\xi_{i})}{2\alpha} \bigl[ \bigl( \mathrm{A}_{1}(\alpha) \bigr)^{1-1/q} \bigl\{ \mathrm{A}_{2}(\alpha) \bigl\vert \psi^{\prime}( \xi_{i}) \bigr\vert ^{q}+\mathrm{A}_{3}(\alpha) \bigl\vert \psi^{\prime}(\xi _{i+1}) \bigr\vert ^{q} \bigr\} ^{1/q} \\ &\quad{} + \bigl(\mathrm{B}_{1}(\alpha) \bigr)^{1-1/q} \bigl\{ \mathrm {B}_{2}(\alpha) \bigl\vert \psi^{\prime}(\xi_{i}) \bigr\vert ^{q} +\mathrm{B}_{3}(\alpha) \bigl\vert \psi^{\prime}(\xi_{i+1}) \bigr\vert ^{q} \bigr\} ^{1/q} \bigr] \end{aligned}$$
holds 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 \(\mathrm{A}_{1}(\alpha)\), \(\mathrm{A}_{2}(\alpha)\), \(\mathrm{A}_{3}(\alpha)\), \(\mathrm{B}_{1}(\alpha)\), \(\mathrm{B}_{2}(\alpha)\) and \(\mathrm{B}_{3}(\alpha)\) are defined as in Theorem 2.5.

5 Conclusion

In this work, we find an identity and several Hermite-Hadamard type inequalities for conformable fractional integrals, present some new inequalities for the arithmetic, logarithmic and generalized logarithmic means of two positive real numbers and provide the error estimations for the trapezoidal formula.

Declarations

Acknowledgements

The research was supported by the Natural Science Foundation of China (Grants Nos. 61673169, 61374086, 11371125, 11401191) and the Tianyuan Special Funds of the National Natural Science Foundation of China (Grant No. 11626101).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Huzhou University, Huzhou, 313000, China
(2)
Department of Mathematics, University of Peshawar, Peshawar, 25000, Pakistan
(3)
College of Engineering and Science, Victoria University, Melbourne, VIC, 8001, Australia

References

  1. Hermite, C: Sur deux limites d’une intégrale définie. Mathesis 3, 82 (1883) Google Scholar
  2. Hadamard, J: Étude sur les propriétés des fonctions entières et en particulier d’une fonction considérée par Riemann. J. Math. Pures Appl. 58, 171-215 (1893) MATHGoogle Scholar
  3. Niculescu, CP, Persson, L-E: Convex Functions and Their Applications. Springer, New York (2006) View ArticleMATHGoogle Scholar
  4. Wang, M-K, Li, Y-M, Chu, Y-M: Inequalities and infinite product formula for Ramanujan generalized modular equation function. Ramanujan J. (2017). doi:10.1007/s11139-017-9888-3 Google Scholar
  5. Wang, M-K, Chu, Y-M: Refinements of transformation inequalities for zero-balanced hypergeometric functions. Acta Math. Sci. 37B(3), 607-622 (2017) MathSciNetView ArticleGoogle Scholar
  6. Adil Khan, M, Khurshid, Y, Ali, T: Hermite-Hadamard inequality for fractional integrals via η-convex functions. Acta Math. Univ. Comen. 86(1), 153-164 (2017) MathSciNetMATHGoogle Scholar
  7. Adil Khan, M, Khurshid, Y, Ali, T, Rehman, N: Inequalities for three times differentiable functions. Punjab Univ. J. Math. 48(2), 35-48 (2016) MathSciNetMATHGoogle Scholar
  8. Chu, Y-M, Adil Khan, M, Khan, TU, Ali, T: Generalizations of Hermite-Hadamard type inequalities for MT-convex functions. J. Nonlinear Sci. Appl. 9(6), 4305-4316 (2016) MathSciNetMATHGoogle Scholar
  9. Wu, Y, Qi, F, Niu, D-W: Integral inequalities of Hermite-Hadamard type for the product of strongly logarithmically convex and other convex functions. Maejo Int. J. Sci. Technol. 9(3), 394-402 (2015) Google Scholar
  10. Noor, MA, Noor, KI, Awan, MU: Hermite-Hadamard inequalities for relative semi-convex functions and applications. Filomat 28(2), 221-230 (2014) MathSciNetView ArticleMATHGoogle Scholar
  11. Bai, R-F, Qi, F, Xi, B-Y: Hermite-Hadamard type inequalities for the m- and \((\alpha, m)\)-logarithmically convex functions. Filomat 27(1), 1-7 (2013) MathSciNetView ArticleMATHGoogle Scholar
  12. Matłoka, M: On some Hadamard-type inequalities for \((h_{1}, h_{2})\)-preinvex functions on the co-ordinates. J. Inequal. Appl. 2013, Article ID 227 (2013) View ArticleMATHGoogle Scholar
  13. Zhang, X-M, Chu, Y-M, Zhang, X-H: The Hermite-Hadamard type inequality of GA-convexity functions. J. Inequal. Appl. 2010, Article ID 507560 (2010) MATHGoogle Scholar
  14. Chu, Y-M, Wang, G-D, Zhang, X-H: Schur convexity and Hadamard’s inequality. Math. Inequal. Appl. 13(4), 725-731 (2010) MathSciNetMATHGoogle Scholar
  15. Bombardelli, M, Varošanec, S: Properties of h-convex functions related to the Hermite-Hadamard-Fejér inequalities. Comput. Math. Appl. 58(9), 1869-1877 (2009) MathSciNetView ArticleMATHGoogle Scholar
  16. Sarikaya, MZ, Saglam, A, Yildirim, H: On some Hadamard-type inequalities for h-convex functons. J. Math. Inequal. 2(3), 335-341 (2008) MathSciNetView ArticleMATHGoogle Scholar
  17. Kirmaci, US, Klaričić Bakula, M, Özdemir, ME, Pečarić, J: Hadamard-type inequalities for s-convex functions. Appl. Math. Comput. 193(1), 26-35 (2007) MathSciNetMATHGoogle Scholar
  18. Noor, MA: On Hadamard integral inequalities involving two log-preinvex functions. JIPAM. J. Inequal. Pure Appl. Math. 8(3), Article ID 75 (2007) MathSciNetMATHGoogle Scholar
  19. Dragomir, SS, McAndrew, A: Refinments of the Hermite-Hadamard inequality for convex functions. JIPAM. J. Inequal. Pure Appl. Math. 6(5), Article ID 140 (2005) MATHGoogle Scholar
  20. Dragomir, SS, Fitzpatrick, S: The Hadamard inequalities for s-convex functions in the second sense. Demonstr. Math. 32(4), 687-696 (1999) MathSciNetMATHGoogle Scholar
  21. Dragomir, SS, Pečarić, JE, Persson, L-E: Some inequalities of Hadamard type. Soochow J. Math. 21(5), 335-341 (1995) MathSciNetMATHGoogle Scholar
  22. Dragomir, SS: Two mappings in connection to Hadamard’s inequalities. J. Math. Anal. Appl. 167(1), 49-56 (1992) MathSciNetView ArticleMATHGoogle Scholar
  23. Dragomir, SS, Agarwal, RP: Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula. Appl. Math. Lett. 11(5), 91-95 (1998) MathSciNetView ArticleMATHGoogle Scholar
  24. Pearce, CEM, Pečarić, JE: Inequalities for differentiable mapping with application to special means and quadrature formulae. Appl. Math. Lett. 13(2), 51-55 (2000) MathSciNetView ArticleMATHGoogle Scholar
  25. Iyiola, OS, Nwaeze, ER: Some new results on the new conformable fractional calculus with application using D’Alambert approach. Prog. Fract. Differ. Appl. 2(2), 115-122 (2016) View ArticleGoogle Scholar
  26. Abdeljawad, T: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57-66 (2015) MathSciNetView ArticleMATHGoogle Scholar
  27. Hammad, MA, Khalil, R: Conformable fractional heat differential equations. Int. J. Pure Appl. Math. 94(2), 215-221 (2014) MATHGoogle Scholar
  28. Hammad, MA, Khalil, R: Abel’s formula and Wronskian for conformable fractional differential equations. Int. J. Differ. Equ. Appl. 13(3), 177-183 (2014) MATHGoogle Scholar
  29. Khalil, R, Al Horani, M, Yousef, A, Sababheh, M: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65-70 (2014) MathSciNetView ArticleMATHGoogle Scholar
  30. Cheng, J-F, Chu, Y-M: On the fractional difference equations of order \((2, q)\). Abstr. Appl. Anal. 2011, Article ID 497259 (2011) MathSciNetMATHGoogle Scholar
  31. Cheng, J-F, Chu, Y-M: Solution to the linear fractional differential equation using Adomian decomposition method. Math. Probl. Eng. 2011, Article ID 587068 (2011) MathSciNetView ArticleMATHGoogle Scholar
  32. Cheng, J-F, Chu, Y-M: Fractional difference equations with real variable. Abstr. Appl. Anal. 2012, Article ID 918529 (2012) MathSciNetMATHGoogle Scholar
  33. Anderson, DR: Taylor’s formula and integral inequalities for conformable fractional derivatives. In: Contributions in Mathematics and Engineering, in Honor of Constantin Carathéodory. Springer, Berlin (2016) Google Scholar

Copyright

© The Author(s) 2017

Advertisement