Skip to main content

Hermite–Hadamard type inequalities for fractional integrals via Green’s function

Abstract

In the article, we establish the left Riemann–Liouville fractional Hermite–Hadamard type inequalities and the generalized Hermite–Hadamard type inequalities by using Green’s function and Jensen’s inequality, and present several new Hermite–Hadamard type inequalities for a class of convex as well as monotone functions.

Introduction

Convexity plays an important role in all the fields of pure and applied mathematics [112]. Many remarkable inequalities have been obtained in the literature by using convexity [1322]. Among the inequalities, the most extensively and intensively attractive inequality in the last decades is the well-known Hermite–Hadamard inequality. This interesting result was obtained by Hermite and Hadamard independently, and it provides an equivalence with the convexity property. This inequality reads as follows: if the function \(\psi:[\alpha_{1},\alpha_{2}]\rightarrow \mathbb{R}\) is convex on \([\alpha_{1},\alpha_{2}]\), then

$$ \psi \biggl(\frac{\alpha_{1}+\alpha_{2}}{2} \biggr)\leq \frac{1}{\alpha_{2}-\alpha_{1}} \int_{\alpha_{1}}^{\alpha_{2}}\psi(x)\,dx \leq \frac{\psi(\alpha_{1})+\psi(\alpha_{2})}{2}. $$
(1.1)

If ψ is a concave function, then the inequalities in (1.1) will hold in reverse directions. The Hermite–Hadamard inequality gives an upper as well as lower estimations for the integral mean of any convex function defined on a closed and bounded interval which involves the endpoints and midpoint of the domain of the function. Also (1.1) provides the necessary and sufficient condition for the function to be convex. There are several applications of this inequality in the geometry of Banach spaces and nonlinear analysis [23, 24]. Some peculiar convex functions can be used in (1.1) to obtain classical inequalities for means. For some comprehensive surveys on various generalizations and developments of (1.1), we recommend [25]. Due to the great importance of this inequality, in the recent years many remarkable varieties of generalizations, refinements, extensions and different versions of Hermite–Hadamard inequality for different classes of convexity, such as preinvex, s-convex, harmonic convex, \(\alpha(x)\)-convex, superquadratic, and co-ordinate convex functions, have been studied in the literature. Also there have been a large number of research papers published on this subject, for interested readers we recommend to read the papers [2637] and some of the references therein.

The following definitions for the left and right side Riemann–Liouville fractional integrals are well known in the literature.

Let \(b_{1}, b_{2}\in\mathbb{R}\) with \(b_{1}< b_{2}\) and \(\psi\in L[b_{1},b_{2}]\). Then the left and right Riemann–Liouville fractional integrals \(J_{b_{1}+}^{\alpha}\psi\) and \(J_{b_{2}-}^{\alpha}\psi\) of order \(\alpha>0\) are defined by

$$ J_{b_{1}+}^{\alpha}\psi(x)=\frac{1}{\Gamma(\alpha)} \int_{b_{1}}^{x}(x-t)^{\alpha-1}\psi(t)\,dt, \quad x>b_{1}, $$

and

$$ J_{b_{2}-}^{\alpha}\psi(x)=\frac{1}{\Gamma(\alpha)} \int_{x}^{b_{2}}(t-x)^{\alpha-1}\psi(t)\,dt, \quad x< b_{2}, $$

respectively, where \(\Gamma(\alpha)\) is the gamma function defined by \(\Gamma(\alpha)=\int_{0}^{\infty}e^{-t}t^{\alpha-1}\,dt\).

In [38], Sarikaya et al. established the Hermite–Hadamard type inequality for fractional integral as follows.

Theorem 1.1

Let \(\psi:[b_{1},b_{2}]\rightarrow\mathbb{R}\) be a positive function with \(0\leq b_{1}< b_{2}\), \(\alpha>0\), and \(\psi\in L[b_{1},b_{2}]\). If ψ is convex on \([b_{1},b_{2}]\), then one has

$$ \psi \biggl(\frac{b_{1}+b_{2}}{2} \biggr)\leq\frac{\Gamma(\alpha+1)}{2(b_{2}-b_{1})^{\alpha}} \bigl[J_{b_{1}+}^{\alpha}\psi(b_{2}) \bigr] +J_{b_{2}-}^{\alpha}\psi(b_{1})\leq\frac{\psi(b_{1})+\psi(b_{2})}{2}. $$

Remark 1.2

In Theorem 1.1, it is not necessary to suppose that ψ is a positive function and \(b_{1},b_{2}\) are positive real numbers. From the definition of left and right Riemann–Liouville fractional integrals, we clearly see that \(b_{1}\) and \(b_{2}\) can be any real numbers such that \(b_{1}< b_{2}\).

The main purpose of this paper is to give a new method to derive the left Riemann–Liouville fractional Hermite–Hadamard type inequalities as given in [39]. In this method we use Green’s function and obtain identities for the difference of the left Riemann–Liouville fractional Hermite–Hadamard inequality, and then we prove that these identities are non-negative. As a consequence, these inequalities provide the generalized Hermite–Hadamard inequality. Also, by using these identities for the class of convex, concave, and monotone functions, we obtain new Hermite–Hadamard type inequalities.

Main results

Let \(b_{1}< b_{2}\). Then the following four new Green’s functions \(G_{i}: [b_{1}, b_{2}]\times [b_{1}, b_{2}]\mapsto \mathbb{R}\) \((i=1, 2, 3, 4)\) are defined by Mehmood et al. in [40]:

$$\begin{aligned} &G_{1}(\lambda, \mu)= \textstyle\begin{cases} b_{1}-\mu , \quad b_{1}\leq \mu\leq\lambda,\\ b_{1}-\lambda , \quad \lambda\leq \mu\leq b_{2}, \end{cases}\displaystyle \end{aligned}$$
(2.1)
$$\begin{aligned} &G_{2}(\lambda, \mu)= \textstyle\begin{cases} \lambda-b_{2}, \quad b_{1}\leq \mu\leq\lambda,\\ \mu-b_{2}, \quad \lambda\leq \mu\leq b_{2}, \end{cases}\displaystyle \end{aligned}$$
(2.2)
$$\begin{aligned} &G_{3}(\lambda, \mu)= \textstyle\begin{cases} \lambda-b_{1}, \quad b_{1}\leq \mu\leq\lambda,\\ \mu-b_{1}, \quad \lambda\leq \mu\leq b_{2}, \end{cases}\displaystyle \end{aligned}$$
(2.3)
$$\begin{aligned} &G_{4}(\lambda, \mu)= \textstyle\begin{cases} b_{2}-\mu, \quad b_{1}\leq \mu\leq\lambda,\\ b_{2}-\lambda, \quad \lambda\leq \mu\leq b_{2}. \end{cases}\displaystyle \end{aligned}$$
(2.4)

In [40], the authors established the following Lemma 2.1, which will be used to establish our main results.

Lemma 2.1

(see [40, Lemma 1])

Let \(b_{1}< b_{2}\) and \(G=G_{1}\) be the Green’s function defined by (2.1). Then

$$ \psi(x)= \psi(b_{1})+(x-b_{1}) \psi'(b_{2})+ \int_{b_{1}}^{b_{2}} G(x,\mu)\psi''( \mu) \,d\mu $$
(2.5)

if \(\psi \in C^{2}([b_{1},b_{2}])\).

Theorem 2.2

Let \(\psi \in C^{2}([b_{1},b_{2}])\) be a convex function. Then the double inequality

$$ \psi \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr) \leq \frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \leq \frac{\alpha\psi(b_{1})+\psi(b_{2})}{\alpha+1} $$
(2.6)

holds for any \(\alpha>0\).

Proof

Substituting \(x= \frac{\alpha b_{1}+b_{2}}{\alpha+1}\) in (2.5), we get

$$ \begin{aligned} & \begin{aligned} \psi \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr)={}&\psi(b_{1})+ \biggl( \frac{\alpha b_{1}+b_{2}}{\alpha+1}-b_{1} \biggr)\psi'(b_{2}) \\ &{}+ \int_{b_{1}}^{b_{2}} G \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1},\mu \biggr) \psi''(\mu) \,d\mu, \end{aligned} \\ &\psi \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr)=\psi(b_{1})+ \frac{(b_{2}-b_{1})\psi'(b_{2})}{\alpha+1}+ \int_{b_{1}}^{b_{2}} G \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1},\mu \biggr) \psi''(\mu) \,d\mu. \end{aligned} $$
(2.7)

Now, multiplying both sides of (2.5) by \(\frac{\alpha(b_{2}-x)^{\alpha-1}}{(b_{2}-b_{1})^{\alpha}}\) and integrating, we get

$$ \begin{aligned} &\frac{\alpha}{(b_{2}-b_{1})^{\alpha}} \int_{b_{1}}^{b_{2}}(b_{2}-x)^{\alpha-1}\psi(x) \,dx \\ &\quad =\frac{\alpha}{(b_{2}-b_{1})^{\alpha}} \biggl[ \int_{b_{1}}^{b_{2}}(b_{2}-x)^{\alpha-1} \psi(b_{1})\,dx+ \int_{b_{1}}^{b_{2}}(b_{2}-x)^{\alpha-1}(x-b_{1}) \psi'(b_{2})\,dx \\ &\qquad {}+ \int_{b_{1}}^{b_{2}} \int_{b_{1}}^{b_{2}}(b_{2}-x)^{\alpha-1}G(x, \mu)\psi''(\mu) \,d\mu \,dx \biggr] \\ &\quad =\frac{\alpha}{(b_{2}-b_{1})^{\alpha}} \biggl[\psi(b_{1})\frac{(b_{2}-b_{1})^{\alpha}}{\alpha}+ \frac{\psi'(b_{2})(b_{2}-b_{1})^{\alpha+1}}{\alpha(\alpha+1)} \\ &\qquad {}+ \int_{b_{1}}^{b_{2}} \int_{b_{1}}^{b_{2}}(b_{2}-x)^{\alpha-1}G(x, \mu)\psi''(\mu) \,d\mu \,dx \biggr], \\ & \begin{aligned} \frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) ={}& \psi(b_{1})+\frac{(b_{2}-b_{1})\psi'(b_{2})}{\alpha+1} \\ &{} +\frac{\alpha}{(b_{2}-b_{1})^{\alpha}} \int_{b_{1}}^{b_{2}} \int_{b_{1}}^{b_{2}}(b_{2}-x)^{\alpha-1}G(x, \mu)\psi''(\mu) \,d\mu \,dx. \end{aligned} \end{aligned} $$
(2.8)

Subtracting (2.8) from (2.7), we obtain

$$ \begin{aligned}[b] &\psi \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr)- \frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha}\psi(b_{2}) \\ &\quad = \int_{b_{1}}^{b_{2}} \biggl[G \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1},\mu \biggr)-\frac{\alpha}{(b_{2}-b_{1})^{\alpha}} \int_{b_{1}}^{b_{2}}(b_{2}-x)^{\alpha-1}G(x, \mu)\,dx \biggr]\psi''(\mu) \,d\mu. \end{aligned} $$
(2.9)

Clearly

$$ \int_{b_{1}}^{b_{2}}(b_{2}-x)^{\alpha-1}G(x, \mu)\,dx=\frac{1}{\alpha(\alpha+1)} \bigl[(b_{2}-\mu)^{\alpha+1}-(b_{2}-b_{1})^{\alpha+1} \bigr]. $$
(2.10)

Also, since

$$ G \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1},\mu \biggr)= \textstyle\begin{cases} b_{1}-\mu, \quad b_{1}\leq \mu\leq\frac{\alpha b_{1}+b_{2}}{\alpha+1}, \\ \frac{b_{1}-b_{2}}{\alpha+1}, \quad \frac{\alpha b_{1}+b_{2}}{\alpha+1}\leq \mu\leq b_{2}. \end{cases} $$
(2.11)

Therefore, if \(b_{1}\leq\mu\leq\frac{\alpha b_{1}+b_{2}}{\alpha+1}\), then from (2.10) and (2.11) we have

$$\begin{aligned} &G \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1},\mu \biggr)-\frac{\alpha}{(b_{2}-b_{1})^{\alpha}} \int_{b_{1}}^{b_{2}}(b_{2}-x)^{\alpha-1}G(x, \mu)\,dx \\ &\quad =b_{1}-\mu-\frac{(b_{2}-\mu)^{\alpha+1}-(b_{2}-b_{1})^{\alpha+1}}{(b_{2}-b_{1})^{\alpha}(\alpha+1)}. \end{aligned}$$

Now, let

$$ f(\mu)=b_{1}-\mu-\frac{(b_{2}-\mu)^{\alpha+1}-(b_{2}-b_{1})^{\alpha+1}}{(b_{2}-b_{1})^{\alpha}(\alpha+1)}. $$

Then

$$ f'(\mu)=-1 + \frac{(b_{2}-\mu)^{\alpha}}{(b_{2}-b_{1})^{\alpha}} \leq 0, $$

which shows that f is decreasing and \(f(b_{1})\leq 0\), hence \(f(\mu)\leq 0\) for all \(\mu\in[b_{1},\frac{\alpha b_{1}+b_{2}}{\alpha+1}]\), i.e.,

$$ G \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1},\mu \biggr)-\frac{\alpha}{(b_{2}-b_{1})^{\alpha}} \int_{b_{1}}^{b_{2}}(b_{2}-x)^{\alpha-1}G(x, \mu)\,dx \leq 0. $$
(2.12)

If \(\frac{\alpha b_{1}+b_{2}}{\alpha+1}\leq\mu\leq b_{2}\), then making use of (2.10) and (2.11) we get

$$ \begin{aligned}[b] &G \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1},\mu \biggr)- \frac{\alpha}{(b_{2}-b_{1})^{\alpha}} \int_{b_{1}}^{b_{2}}(b_{2}-x)^{\alpha-1}G(x, \mu)\,dx \\ &\quad =\frac{b_{1}-b_{2}}{\alpha+1}-\frac{\alpha}{(b_{2}-b_{1})^{\alpha}} \biggl[\frac{(b_{2}-\mu)^{\alpha+1}-(b_{2}-b_{1})^{\alpha+1}}{\alpha(\alpha+1)} \biggr] \\ &\quad =-\frac{(b_{2}-b_{1})}{(\alpha+1)}-\frac{(b_{2}-\mu)^{\alpha+1}-(b_{2}-b_{1})^{\alpha+1}}{(b_{2}-b_{1})^{\alpha}(\alpha+1)} \\ &\quad =\frac{-(b_{2}-b_{1})^{\alpha+1}-(b_{2}-\mu)^{\alpha+1}+(b_{2}-b_{1})^{\alpha+1}}{(\alpha+1)(b_{2}-b_{1})^{\alpha}} \\ &\quad =\frac{-(b_{2}-\mu)^{\alpha+1}}{(\alpha+1)(b_{2}-b_{1})^{\alpha}} \leq 0. \end{aligned} $$
(2.13)

Since ψ is convex, therefore, \(\psi''(\mu)\geq\) 0 and so by using (2.12) and (2.13) in (2.9), we deduce

$$ \psi \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr)\leq\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}), $$

which is the first inequality of (2.6).

Next, we prove the second inequality of (2.6). Let \(x=b_{2}\) in (2.5), then we have

$$ \psi(b_{2})=\psi(b_{1})+(b_{2}-b_{1}) \psi'(b_{2})+ \int_{b_{1}}^{b_{2}}G(b_{2},\mu) \psi''(\mu)\,d\mu. $$

Adding \(\alpha\psi(b_{1})\) on both sides and then dividing by \((\alpha+1)\), we get

$$ \frac{\alpha\psi(b_{1})+\psi(b_{2})}{\alpha+1}=\psi(b_{1})+\frac{(b_{2}-b_{1})\psi'(b_{2})}{\alpha+1}+ \frac{1}{\alpha+1} \int_{b_{1}}^{b_{2}}G(b_{2},\mu) \psi''(\mu)\,d\mu. $$
(2.14)

Subtracting (2.8) from (2.14), we obtain

$$ \begin{aligned} &\frac{\alpha\psi(b_{1})+\psi(b_{2})}{\alpha+1}-\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \\ &\quad = \int_{b_{1}}^{b_{2}} \biggl[\frac{G(b_{2},\mu)}{\alpha+1}- \frac{\alpha}{(b_{2}-b_{1})^{\alpha}} \int_{b_{1}}^{b_{2}}(b_{2}-x)^{\alpha-1}G(x, \mu)\,dx \biggr]\psi''(\mu)\,d\mu. \end{aligned} $$
(2.15)

Using the Green’s function and (2.10), we obtain

$$ \begin{aligned}[b] &\frac{G(b_{2},\mu)}{\alpha+1}-\frac{\alpha}{(b_{2}-b_{1})^{\alpha}} \biggl[ \frac{(b_{2}-\mu)^{\alpha+1}-(b_{2}-b_{1})^{\alpha+1}}{\alpha(\alpha+1)} \biggr] \\ &\quad =\frac{b_{1}-\mu}{\alpha+1}-\frac{(b_{2}-\mu)^{\alpha+1}-(b_{2}-b_{1})^{\alpha+1}}{(\alpha+1)(b_{2}-b_{1})^{\alpha}} \\ &\quad =\frac{(b_{1}-\mu)(b_{2}-b_{1})^{\alpha}+(b_{2}-b_{1})^{\alpha+1}-(b_{2}-\mu)^{\alpha+1}}{(\alpha+1)(b_{2}-b_{1})^{\alpha}} \\ &\quad =\frac{(b_{2}-b_{1})^{\alpha}(b_{2}-\mu)-(b_{2}-\mu)^{\alpha+1}}{(\alpha+1)(b_{2}-b_{1})^{\alpha}} \\ &\quad =\frac{(b_{2}-\mu)[(b_{2}-b_{1})^{\alpha}-(b_{2}-\mu)^{\alpha}]}{(\alpha+1)(b_{2}-b_{1})^{\alpha}} \geq 0 \end{aligned} $$
(2.16)

for all \(b_{1}\leq\mu\leq b_{2}\).

Now, using the convexity of ψ and (2.16) in (2.15), we get

$$ \frac{\alpha\psi(b_{1})+\psi(b_{2})}{\alpha+1}\geq\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}), $$

which is the second inequality of (2.6). □

Next, we present new Hermite–Hadamard type inequalities for the class of monotone and convex functions.

Theorem 2.3

Let \(\psi \in C^{2}([b_{1},b_{2}])\) and \(\alpha>0\). Then the following statements are true:

  1. (i)

    If \(|\psi''|\) is an increasing function, then

    $$ \biggl\vert \frac{\alpha\psi(b_{1})+\psi(b_{2})}{\alpha+1}-\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \biggr\vert \leq\frac{ \vert \psi''(b_{2}) \vert \alpha(b_{2}-b_{1})^{2}}{2(\alpha+1)(\alpha+2)}. $$
    (2.17)
  2. (ii)

    If \(|\psi''|\) is a decreasing function, then

    $$ \biggl\vert \frac{\alpha\psi(b_{1})+\psi(b_{2})}{\alpha+1}-\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \biggr\vert \leq\frac{ \vert \psi''(b_{1}) \vert \alpha(b_{2}-b_{1})^{2}}{2(\alpha+1)(\alpha+2)}. $$
  3. (iii)

    If \(|\psi''|\) is a convex function, then

    $$\begin{aligned} \biggl\vert \frac{\alpha\psi(b_{1})+\psi(b_{2})}{\alpha+1}-\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \biggr\vert \leq\frac{\max\{ \vert \psi''(b_{1}) \vert , \vert \psi''(b_{2}) \vert \}\alpha(b_{2}-b_{1})^{2}}{2(\alpha+1)(\alpha+2)}. \end{aligned}$$

Proof

(i) It follows from (2.15) that

$$ \begin{aligned}[b] & \biggl\vert \frac{\alpha\psi(b_{1})+\psi(b_{2})}{\alpha+1}- \frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \biggr\vert \\ &\quad \leq\frac{1}{(\alpha+1)(b_{2}-b_{1})^{\alpha}} \int_{b_{1}}^{b_{2}} \bigl[(b_{2}-b_{1})^{\alpha}(b_{2}- \mu)-(b_{2}-\mu)^{\alpha+1} \bigr] \bigl\vert \psi''( \mu) \bigr\vert \,d\mu. \end{aligned} $$
(2.18)

Since \((b_{2}-b_{1})^{\alpha}(b_{2}-\mu)-(b_{2}-\mu)^{\alpha+1}\geq0\) and \(|\psi''|\) is an increasing function, therefore we have

$$\begin{aligned} & \biggl\vert \frac{\alpha\psi(b_{1})+\psi(b_{2})}{\alpha+1}-\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \biggr\vert \\ &\quad \leq\frac{ \vert \psi''(b_{2}) \vert }{(\alpha+1)(b_{2}-b_{1})^{\alpha}} \int_{b_{1}}^{b_{2}} \bigl[(b_{2}-b_{1})^{\alpha}(b_{2}- \mu)-(b_{2}-\mu)^{\alpha+1} \bigr]\,d\mu \\ &\quad =\frac{ \vert \psi''(b_{2}) \vert }{(\alpha+1)(b_{2}-b_{1})^{\alpha}} \biggl[(b_{2}-b_{1})^{\alpha}\frac{(b_{2}-b_{1})^{2}}{2}-\frac{(b_{2}-b_{1})^{\alpha+2}}{\alpha+2} \biggr] \\ &\quad =\frac{ \vert \psi''(b_{2}) \vert \alpha(b_{2}-b_{1})^{2}}{2(\alpha+1)(\alpha+2)}, \end{aligned}$$

which is inequality (2.17).

Part (ii) can be proved in a similar way, we omit the details.

For part (iii), making use of (2.18) and the fact that every convex function ψ defined on the interval \([b_{1},b_{2}]\) is bounded above by \(\max\{\psi(b_{1}),\psi(b_{2})\}\), we get

$$\begin{aligned} & \biggl\vert \frac{\alpha\psi(b_{1})+\psi(b_{2})}{\alpha+1}-\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \biggr\vert \\ &\quad \leq\frac{\max\{ \vert \psi''(b_{1}) \vert , \vert \psi''(b_{2}) \vert \}}{(\alpha+1)(b_{2}-b_{1})^{\alpha}} \int_{b_{1}}^{b_{2}} \bigl[(b_{2}-b_{1})^{\alpha}(b_{2}- \mu)-(b_{2}-\mu)^{\alpha+1} \bigr]\,d\mu \\ &\quad =\frac{\max\{ \vert \psi''(b_{1}) \vert , \vert \psi''(b_{2}) \vert \}\alpha(b_{2}-b_{1})^{2}}{2(\alpha+1)(\alpha+2)}. \end{aligned}$$

 □

Remark 2.4

Let \(\alpha=1\). Then Theorem 2.3 leads to

$$\begin{aligned} & \biggl\vert \frac{\psi(b_{1})+\psi(b_{2})}{2}-\frac{1}{b_{2}-b_{1}} \int_{b_{1}}^{b_{2}}\psi(t)\,dt \biggr\vert \leq \frac{ \vert \psi''(b_{2}) \vert (b_{2}-b_{1})^{2}}{12}, \\ & \biggl\vert \frac{\psi(b_{1})+\psi(b_{2})}{2}-\frac{1}{b_{2}-b_{1}} \int_{b_{1}}^{b_{2}}\psi(t)\,dt \biggr\vert \leq \frac{ \vert \psi''(b_{1}) \vert (b_{2}-b_{1})^{2}}{12}, \\ & \biggl\vert \frac{\psi(b_{1})+\psi(b_{2})}{2}-\frac{1}{b_{2}-b_{1}} \int_{b_{1}}^{b_{2}}\psi(t)\,dt \biggr\vert \leq \frac{\max\{ \vert \psi''(b_{1}) \vert , \vert \psi''(b_{2}) \vert \}(b_{2}-b_{1})^{2}}{12}. \end{aligned}$$

Theorem 2.5

Let \(\psi \in C^{2}([b_{1},b_{2}])\) and \(\alpha>0\). Then the following statements are true:

  1. (i)

    If \(|\psi''|\) is an increasing function, then

    $$\begin{aligned} & \biggl\vert \psi \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr)-\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \biggr\vert \\ &\quad \leq\frac{(b_{2}-b_{1})^{2}}{(\alpha+1)^{\alpha+3}(\alpha+2)} \\ &\qquad {}\times\biggl[ \biggl\vert \psi'' \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr) \biggr\vert \biggl(\frac{\alpha[(\alpha+1)^{\alpha+1}-2\alpha^{\alpha+1}]}{2} \biggr)+ \bigl\vert \psi''(b_{2}) \bigr\vert \alpha^{\alpha+2} \biggr]. \end{aligned}$$
  2. (ii)

    If \(|\psi''|\) is a decreasing function, then

    $$\begin{aligned} &\biggl\vert \psi \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr)-\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \biggr\vert \\ &\quad \leq\frac{(b_{2}-b_{1})^{2}}{(\alpha+1)^{\alpha+3}(\alpha+2)} \\ &\qquad {}\times\biggl[ \bigl\vert \psi''(b_{1}) \bigr\vert \biggl(\frac{\alpha[(\alpha+1)^{\alpha+1}-2\alpha^{\alpha+1}]}{2} \biggr)+ \biggl\vert \psi'' \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr) \biggr\vert \alpha^{\alpha+2} \biggr]. \end{aligned}$$
  3. (iii)

    If \(|\psi''|\) is a convex function, then

    $$\begin{aligned} &\biggl\vert \psi \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr)-\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \biggr\vert \\ &\quad \leq\frac{(b_{2}-b_{1})^{2}}{(\alpha+1)^{\alpha+3}(\alpha+2)} \\ &\qquad {}\times \biggl[\max \biggl\{ \bigl\vert \psi''(b_{1}) \bigr\vert , \biggl\vert \psi'' \biggl( \frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr) \biggr\vert \biggr\} \biggl(\frac{\alpha[(\alpha+1)^{\alpha+1}-2\alpha^{\alpha+1}]}{2} \biggr) \\ &\qquad {}+\max \biggl\{ \biggl\vert \psi'' \biggl( \frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr) \biggr\vert , \bigl\vert \psi''(b_{2}) \bigr\vert \biggr\} \alpha^{\alpha+2} \biggr]. \end{aligned}$$

Proof

(i) It follows from (2.9) that

$$\begin{aligned} &\psi \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr)- \frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \\ &\quad = \int_{b_{1}}^{\frac{\alpha b_{1}+b_{2}}{\alpha+1}} \biggl[G \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1},\mu \biggr)-\frac{\alpha}{(b_{2}-b_{1})^{\alpha}} \int_{b_{1}}^{b_{2}}(b_{2}-x)^{\alpha-1}G(x, \mu)\,dx \biggr]\psi''(\mu) \,d\mu \\ &\qquad {}+ \int_{\frac{\alpha b_{1}+b_{2}}{\alpha+1}}^{b_{2}} \biggl[G \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1},\mu \biggr)-\frac{\alpha}{(b_{2}-b_{1})^{\alpha}} \int_{b_{1}}^{b_{2}}(b_{2}-x)^{\alpha-1}G(x, \mu)\,dx \biggr]\psi''(\mu) \,d\mu \\ &\quad =-\frac{1}{(\alpha+1)(b_{2}-b_{1})^{\alpha}} \biggl[ \int_{b_{1}}^{\frac{\alpha b_{1}+b_{2}}{\alpha+1}} \bigl\{ (b_{2}- \mu)^{\alpha+1} -(b_{2}-b_{1})^{\alpha+1} \\ &\qquad {}-(b_{1}-\mu) (b_{2}-b_{1})^{\alpha}( \alpha+1) \bigr\} \psi''(\mu)\,d\mu + \int_{\frac{\alpha b_{1}+b_{2}}{\alpha+1}}^{b_{2}}(b_{2}-\mu)^{\alpha+1} \psi''(\mu)\,d\mu \biggr]. \end{aligned}$$
(2.19)

Taking the absolute function and using the triangular inequality, we get

$$\begin{aligned} & \biggl\vert \psi \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr)-\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \biggr\vert \\ &\quad \leq\frac{1}{(\alpha+1)(b_{2}-b_{1})^{\alpha}} \biggl[ \int_{b_{1}}^{\frac{\alpha b_{1}+b_{2}}{\alpha+1}} \bigl\{ (b_{2}- \mu)^{\alpha+1}-(b_{2}-b_{1})^{\alpha+1} \\ &\qquad {}-(b_{1}-\mu) (b_{2}-b_{1})^{\alpha}( \alpha+1) \bigr\} \bigl\vert \psi''(\mu) \bigr\vert \,d \mu + \int_{\frac{\alpha b_{1}+b_{2}}{\alpha+1}}^{b_{2}}(b_{2}-\mu)^{\alpha+1} \bigl\vert \psi''(\mu) \bigr\vert \,d\mu \biggr] \\ &\quad \leq\frac{1}{(\alpha+1)(b_{2}-b_{1})^{\alpha}} \biggl[ \biggl\vert \psi'' \biggl( \frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr) \biggr\vert \int_{b_{1}}^{\frac{\alpha b_{1}+b_{2}}{\alpha+1}} \bigl\{ (b_{2}- \mu)^{\alpha+1}-(b_{2}-b_{1})^{\alpha+1} \\ &\qquad {}-(b_{1}-\mu) (b_{2}-b_{1})^{\alpha}( \alpha+1) \bigr\} \,d\mu+ \bigl\vert \psi''(b_{2}) \bigr\vert \int_{\frac{\alpha b_{1}+b_{2}}{\alpha+1}}^{b_{2}}(b_{2}-\mu)^{\alpha+1}\,d \mu \biggr] \\ &\quad =\frac{1}{(\alpha+1)(b_{2}-b_{1})^{\alpha}} \biggl[ \biggl\vert \psi'' \biggl( \frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr) \biggr\vert (b_{2}-b_{1})^{\alpha+2} \biggl\{ \frac{-\alpha^{\alpha+2}}{(\alpha+1)^{\alpha+2}(\alpha+2)} \\ &\qquad {}+\frac{1}{(\alpha+2)} -\frac{1}{(\alpha+1)}+\frac{1}{2(\alpha+1)} \biggr\} + \bigl\vert \psi''(b_{2}) \bigr\vert (b_{2}-b_{1})^{\alpha+2} \biggl\{ \frac{\alpha^{\alpha+2}}{(\alpha+1)^{\alpha+2}(\alpha+2)} \biggr\} \biggr] \\ &\quad =\frac{(b_{2}-b_{1})^{2}}{(\alpha+1)^{\alpha+3}(\alpha+2)} \biggl[ \biggl\vert \psi'' \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr) \biggr\vert \biggl\{ \frac{\alpha[(\alpha+1)^{\alpha+1}-2\alpha^{\alpha+1}]}{2} \biggr\} + \bigl\vert \psi''(b_{2}) \bigr\vert \alpha^{\alpha+2} \biggr]. \end{aligned}$$

Part (ii) can be proved by using the same procedure.

Next, we prove part (iii). We clearly see that

$$\begin{aligned} & \biggl\vert \psi \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr)-\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \biggr\vert \\ &\quad \leq\frac{1}{(\alpha+1)(b_{2}-b_{1})^{\alpha}} \biggl[ \int_{b_{1}}^{\frac{\alpha b_{1}+b_{2}}{\alpha+1}} \bigl\{ (b_{2}- \mu)^{\alpha+1} -(b_{2}-b_{1})^{\alpha+1} \\ &\qquad {}-(b_{1}-\mu) (b_{2}-b_{1})^{\alpha}( \alpha+1) \bigr\} \bigl\vert \psi''(\mu) \bigr\vert \,d \mu + \int_{\frac{\alpha b_{1}+b_{2}}{\alpha+1}}^{b_{2}}(b_{2}-\mu)^{\alpha+1} \bigl\vert \psi''(\mu) \bigr\vert \,d\mu \biggr]. \end{aligned}$$

Since every convex function ψ defined on an interval \([b_{1},b_{2}]\) is bounded above by \(\max\{\psi(b_{1}),\psi(b_{2})\}\). Therefore, we have

$$\begin{aligned} & \biggl\vert \psi \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr)-\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \biggr\vert \\ &\quad \leq\frac{1}{(\alpha+1)(b_{2}-b_{1})^{\alpha}} \biggl[\max \biggl\{ \bigl\vert \psi''(b_{1}) \bigr\vert , \biggl\vert \psi'' \biggl( \frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr) \biggr\vert \biggr\} , \\ & \int_{b_{1}}^{\frac{\alpha b_{1}+b_{2}}{\alpha+1}} \bigl\{ (b_{2}- \mu)^{\alpha+1}-(b_{2}-b_{1})^{\alpha+1}-(b_{1}- \mu) (b_{2}-b_{1})^{\alpha}(\alpha+1) \bigr\} \,d\mu \\ &\qquad {}+\max \biggl\{ \biggl\vert \psi'' \biggl( \frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr) \biggr\vert , \bigl\vert \psi''(b_{2}) \bigr\vert \biggr\} \int_{\frac{\alpha b_{1}+b_{2}}{\alpha+1}}^{b_{2}}(b_{2}-\mu)^{\alpha+1}\,d \mu \biggr] \\ &\quad =\frac{(b_{2}-b_{1})^{2}}{(\alpha+1)^{\alpha+3}(\alpha+2)} \biggl[\max \biggl\{ \bigl\vert \psi''(b_{1}) \bigr\vert , \biggl\vert \psi'' \biggl( \frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr) \biggr\vert \biggr\} \biggl(\frac{\alpha[(\alpha+1)^{\alpha+1}-2\alpha^{\alpha+1}]}{2} \biggr) \\ & \qquad {}+\max \biggl\{ \biggl\vert \psi'' \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr) \biggr\vert , \bigl\vert \psi''(b_{2}) \bigr\vert \biggr\} \bigl(\alpha^{\alpha+2} \bigr) \biggr], \end{aligned}$$

which is our required inequality. □

Remark 2.6

Let \(\alpha=1\). Then Theorem 2.5 leads to

$$\begin{aligned} & \biggl\vert \psi \biggl(\frac{b_{1}+b_{2}}{2} \biggr)-\frac{1}{b_{2}-b_{1}} \int_{b_{1}}^{b_{2}}\psi(t)\,dt \biggr\vert \leq \frac{(b_{2}-b_{1})^{2}}{48} \biggl[ \biggl\vert \psi'' \biggl( \frac{b_{1}+b_{2}}{2} \biggr) \biggr\vert + \bigl\vert \psi''(b_{2}) \bigr\vert \biggr], \\ & \biggl\vert \psi \biggl(\frac{b_{1}+b_{2}}{2} \biggr)-\frac{1}{b_{2}-b_{1}} \int_{b_{1}}^{b_{2}}\psi(t)\,dt \biggr\vert \leq \frac{(b_{2}-b_{1})^{2}}{48} \biggl[ \bigl\vert \psi''(b_{1}) \bigr\vert + \biggl\vert \psi'' \biggl( \frac{b_{1}+b_{2}}{2} \biggr) \biggr\vert \biggr], \\ & \biggl\vert \psi \biggl(\frac{b_{1}+b_{2}}{2} \biggr)-\frac{1}{b_{2}-b_{1}} \int_{b_{1}}^{b_{2}}\psi(t)\,dt \biggr\vert \\ &\quad \leq\frac{(b_{2}-b_{1})^{2}}{48} \biggl[\max \biggl\{ \bigl\vert \psi''(b_{1}) \bigr\vert , \biggl\vert \psi'' \biggl( \frac{b_{1}+b_{2}}{2} \biggr) \biggr\vert \biggr\} +\max \biggl\{ \biggl\vert \psi'' \biggl( \frac{b_{1}+b_{2}}{2} \biggr) \biggr\vert , \bigl\vert \psi''(b_{2}) \bigr\vert \biggr\} \biggr]. \end{aligned}$$

Theorem 2.7

Let \(\psi \in C^{2}([b_{1},b_{2}])\) and \(|\psi''|\) be a convex function. Then the inequality

$$\begin{aligned} & \biggl\vert \psi \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr)-\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \biggr\vert \\ &\quad \leq\frac{(b_{2}-b_{1})^{2}}{6(\alpha+1)^{3}(\alpha+3)} \biggl[ \bigl\vert \psi''(b_{1}) \bigr\vert \bigl(9\alpha^{2}+23\alpha+12 \bigr)+\frac{ \vert \psi''(b_{2}) \vert (7\alpha^{2}+17\alpha+12)}{\alpha+2} \biggr] \end{aligned}$$

holds for any \(\alpha>0\).

Proof

It follows from (2.19) that

$$\begin{aligned} &\psi \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr)-\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \\ &\quad =\frac{1}{(\alpha+1)(b_{2}-b_{1})^{\alpha}} \biggl[ \int_{b_{1}}^{\frac{\alpha b_{1}+b_{2}}{\alpha+1}} \bigl\{ (b_{1}-\mu) (b_{2}-b_{1})^{\alpha}(\alpha+1) \\ &\qquad {}-(b_{2}-\mu)^{\alpha+1} +(b_{2}-b_{1})^{\alpha+1} \bigr\} \psi''(\mu)\,d\mu - \int_{\frac{\alpha b_{1}+b_{2}}{\alpha+1}}^{b_{2}}(b_{2}-\mu)^{\alpha+1} \psi''(\mu)\,d\mu \biggr]. \end{aligned}$$

Let \(t\in[0,1]\) and \(\mu=tb_{1}+(1-t)b_{2}\). Then

$$ \begin{aligned}[b] &\psi \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr)- \frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \\ &\quad =\frac{1}{(\alpha+1)(b_{2}-b_{1})^{\alpha}} \biggl[ \int_{1}^{\frac{\alpha}{\alpha+1}} \bigl\{ \bigl[b_{1}-tb_{1}-(1-t)b_{2} \bigr](b_{2}-b_{1})^{\alpha}(\alpha+1) \\ &\qquad {}- \bigl[b_{2}-tb_{1}-(1-t)b_{2} \bigr]^{\alpha+1}+(b_{2}-b_{1})^{\alpha+1} \bigr\} \psi'' \bigl(tb_{1}+(1-t)b_{2} \bigr) (b_{1}-b_{2})\,dt \\ &\qquad {}- \int_{\frac{\alpha}{\alpha+1}}^{0} \bigl[b_{2}-tb_{1}-(1-t)b_{2} \bigr]^{\alpha+1}\psi'' \bigl(tb_{1}+(1-t)b_{2} \bigr) (b_{1}-b_{2})\,dt \biggr] \\ &\quad =\frac{-(b_{2}-b_{1})^{\alpha+2}}{(\alpha+1)(b_{2}-b_{1})^{\alpha}} \biggl[ \int_{1}^{\frac{\alpha}{\alpha+1}} \bigl\{ (-1+t) (\alpha+1)+1 \bigr\} \psi'' \bigl(tb_{1}+(1-t)b_{2} \bigr) \,dt \\ &\qquad {}- \int_{1}^{\frac{\alpha}{\alpha+1}}t^{\alpha+1}\psi'' \bigl(tb_{1}+(1-t)b_{2} \bigr)\,dt+ \int_{0}^{\frac{\alpha}{\alpha+1}}t^{\alpha+1}\psi'' \bigl(tb_{1}+(1-t)b_{2} \bigr)\,dt \biggr] \\ &\quad =\frac{(b_{2}-b_{1})^{2}}{\alpha+1} \biggl[ \int_{\frac{\alpha}{\alpha+1}}^{1}(\alpha t+t-\alpha) \psi'' \bigl(tb_{1}+(1-t)b_{2} \bigr) \,dt\\ &\qquad {} - \int_{0}^{1}t^{\alpha+1}\psi'' \bigl(tb_{1}+(1-t)b_{2} \bigr)\,dt \biggr]. \end{aligned} $$
(2.20)

Taking absolute on both sides and using the convexity of \(|\psi''|\), we get

$$\begin{aligned} & \biggl\vert \psi \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr)-\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \biggr\vert \\ &\quad \leq\frac{(b_{2}-b_{1})^{2}}{\alpha+1} \biggl[ \int_{\frac{\alpha}{\alpha+1}}^{1}(\alpha t+t-\alpha) \bigl\vert \psi'' \bigl(tb_{1}+(1-t)b_{2} \bigr) \bigr\vert \,dt\\ &\qquad {} + \int_{0}^{1}t^{\alpha+1} \bigl\vert \psi'' \bigl(tb_{1}+(1-t)b_{2} \bigr) \bigr\vert \,dt \biggr] \\ &\quad \leq\frac{(b_{2}-b_{1})^{2}}{\alpha+1} \biggl[ \int_{\frac{\alpha}{\alpha+1}}^{1} \bigl(t(\alpha+1)-\alpha \bigr) \bigl[t \bigl\vert \psi''(b_{1}) \bigr\vert +(1-t) \bigl\vert \psi''(b_{2}) \bigr\vert \bigr]\,dt \\ &\qquad {}+ \int_{0}^{1}t^{\alpha+1} \bigl[t \bigl\vert \psi''(b_{1}) \bigr\vert +(1-t) \bigl\vert \psi''(b_{2}) \bigr\vert \bigr]\,dt \biggr] \\ &\quad =\frac{(b_{2}-b_{1})^{2}}{\alpha+1} \biggl[\frac{ \vert \psi''(b_{1}) \vert (3\alpha+2)}{6(\alpha+1)^{2}}+\frac{ \vert \psi''(b_{2}) \vert }{6(\alpha+1)^{2}} + \frac{ \vert \psi''(b_{1}) \vert }{\alpha+3}+\frac{ \vert \psi''(b_{2}) \vert }{(\alpha+2)(\alpha+3)} \biggr] \\ &\quad =\frac{(b_{2}-b_{1})^{2}}{6(\alpha+1)^{3}(\alpha+3)} \biggl[ \bigl\vert \psi''(b_{1}) \bigr\vert \bigl(9\alpha^{2}+23\alpha+12 \bigr)+\frac{ \vert \psi''(b_{2}) \vert (7\alpha^{2}+17\alpha+12)}{\alpha+2} \biggr], \end{aligned}$$

which completes the proof. □

Remark 2.8

In Theorem 2.7, if we take \(\alpha=1\), then we obtain

$$ \biggl\vert \psi \biggl(\frac{b_{1}+b_{2}}{2} \biggr)-\frac{1}{b_{2}-b_{1}} \int_{b_{1}}^{b_{2}}\psi(t)\,dt \biggr\vert \leq \frac{(b_{2}-b_{1})^{2}}{48} \bigl[11|\psi''(b_{1})+3 \bigl\vert \psi''(b_{2}) \bigr\vert \bigr]. $$

Theorem 2.9

Let \(\psi \in C^{2}([b_{1},b_{2}])\) and \(|\psi''|\) be a convex function. Then the inequality

$$\begin{aligned} & \biggl\vert \frac{\alpha\psi(b_{1})+\psi(b_{2})}{\alpha+1}-\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \biggr\vert \\ &\quad \leq\frac{\alpha(b_{2}-b_{1})^{2}}{3(\alpha+1)(\alpha+3)} \biggl[ \bigl\vert \psi''(b_{1}) \bigr\vert + \bigl\vert \psi''(b_{2}) \bigr\vert \biggl(\frac{\alpha+5}{2(\alpha+2)} \biggr) \biggr] \end{aligned}$$

holds for any \(\alpha>0\).

Proof

From (2.15), we have

$$\begin{aligned} &\frac{\alpha\psi(b_{1})+\psi(b_{2})}{\alpha+1}-\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \\ &\quad =\frac{1}{(\alpha+1)(b_{2}-b_{1})^{\alpha}} \int_{b_{1}}^{b_{2}} \bigl[(b_{2}-b_{1})^{\alpha}(b_{2}- \mu)-(b_{2}-\mu)^{\alpha+1} \bigr]\psi''( \mu)\,d\mu. \end{aligned}$$

Let \(t\in[0,1]\) and \(\mu=tb_{1}+(1-t)b_{2}\). Then we obtain

$$ \begin{aligned}[b] &\frac{\alpha\psi(b_{1})+\psi(b_{2})}{\alpha+1}-\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \\ &\quad =\frac{b_{2}-b_{1}}{(\alpha+1)(b_{2}-b_{1})^{\alpha}}\\ &\qquad {}\times \int_{0}^{1} \bigl\{ (b_{2}-b_{1})^{\alpha}\bigl[t(b_{2}-b_{1}) \bigr]- \bigl[t(b_{2}-b_{1}) \bigr]^{\alpha+1} \bigr\} \psi'' \bigl(tb_{1}+(1-t)b_{2} \bigr)\,dt \\ &\quad =\frac{(b_{2}-b_{1})^{2}}{\alpha+1} \int_{0}^{1} \bigl(t-t^{\alpha+1} \bigr) \psi'' \bigl(tb_{1}+(1-t)b_{2} \bigr) \,dt. \end{aligned} $$
(2.21)

Taking absolute on both sides and using the convexity of \(|\psi''|\), we get

$$\begin{aligned} & \biggl\vert \frac{\alpha\psi(b_{1})+\psi(b_{2})}{\alpha+1}-\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \biggr\vert \\ &\quad \leq\frac{(b_{2}-b_{1})^{2}}{\alpha+1} \int_{0}^{1} \bigl(t-t^{\alpha+1} \bigr) \bigl\vert \psi'' \bigl(tb_{1}+(1-t)b_{2} \bigr) \bigr\vert \,dt \\ &\quad \leq\frac{(b_{2}-b_{1})^{2}}{\alpha+1} \int_{0}^{1} \bigl(t-t^{\alpha+1} \bigr) \bigl[t \bigl\vert \psi''(b_{1}) \bigr\vert +(1-t) \bigl\vert \psi''(b_{2}) \bigr\vert \bigr]\,dt \\ &\quad =\frac{(b_{2}-b_{1})^{2}}{\alpha+1} \biggl[ \bigl\vert \psi''(b_{1}) \bigr\vert \biggl(\frac{\alpha}{3(\alpha+3)} \biggr) + \bigl\vert \psi''(b_{2}) \bigr\vert \biggl( \frac{\alpha^{2}+5\alpha}{6(\alpha+2)(\alpha+3)} \biggr) \biggr] \\ &\quad =\frac{\alpha(b_{2}-b_{1})^{2}}{3(\alpha+1)(\alpha+3)} \biggl[ \bigl\vert \psi''(b_{1}) \bigr\vert + \bigl\vert \psi''(b_{2}) \bigr\vert \biggl(\frac{\alpha+5}{2(\alpha+2)} \biggr) \biggr], \end{aligned}$$

which is our required inequality. □

Remark 2.10

In Theorem 2.9, if we take \(\alpha=1\), then we obtain

$$ \biggl\vert \frac{\psi(b_{1})+\psi(b_{2})}{2}-\frac{1}{b_{2}-b_{1}} \int_{b_{1}}^{b_{2}}\psi(t)\,dt \biggr\vert \leq \frac{(b_{2}-b_{1})^{2}}{24} \bigl[ \bigl\vert \psi''(b_{1})+| \psi''(b_{2}) \bigr\vert \bigr]. $$

Theorem 2.11

Let \(\psi \in C^{2}([b_{1},b_{2}])\) and \(|\psi''|\) be a concave function. Then

$$\begin{aligned} & \biggl\vert \psi \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr)-\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \biggr\vert \\ &\quad \leq\frac{(b_{2}-b_{1})^{2}}{\alpha+1} \biggl[\frac{1}{2(\alpha+1)} \biggl\vert \psi'' \biggl(\frac{3\alpha b_{1}+2b_{1}+b_{2}}{3(\alpha+1)} \biggr) \biggr\vert + \frac{1}{\alpha+2} \biggl\vert \psi'' \biggl( \frac{\alpha b_{1}+2b_{1}+b_{2}}{\alpha+3} \biggr) \biggr\vert \biggr] \end{aligned}$$

for any \(\alpha>0\).

Proof

It follows from (2.20) that

$$\begin{aligned} &\psi \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr)-\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \\ &\quad =\frac{(b_{2}-b_{1})^{2}}{\alpha+1}\\ &\qquad {}\times \biggl[ \int_{\frac{\alpha}{\alpha+1}}^{1} \bigl\{ t(\alpha+1)-\alpha \bigr\} \psi'' \bigl(tb_{1}+(1-t)b_{2} \bigr) \,dt - \int_{0}^{1}t^{\alpha+1}\psi'' \bigl(tb_{1}+(1-t)b_{2} \bigr)\,dt \biggr]. \end{aligned}$$

Taking absolute on both sides and using Jensen’s integral inequality, we get

$$\begin{aligned} & \biggl\vert \psi \biggl(\frac{\alpha b_{1}+b_{2}}{\alpha+1} \biggr)-\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \biggr\vert \\ &\quad \leq\frac{(b_{2}-b_{1})^{2}}{\alpha+1} \biggl[ \int_{\frac{\alpha}{\alpha+1}}^{1} \bigl\{ t(\alpha+1)-\alpha \bigr\} \,dt \biggl\vert \psi'' \biggl(\frac{\int_{\frac{\alpha}{\alpha+1}}^{1}\{t(\alpha+1)-\alpha\}\{tb_{1}+(1-t)b_{2}\}\,dt}{ \int_{\frac{\alpha}{\alpha+1}}^{1}\{t(\alpha+1)-\alpha\}\,dt} \biggr) \biggr\vert \\ &\qquad {}+ \int_{0}^{1}t^{\alpha+1}\,dt \biggl\vert \psi'' \biggl(\frac{\int_{0}^{1}t^{\alpha+1}\{tb_{1}+(1-t)b_{2}\}\,dt}{\int_{0}^{1}t^{\alpha+1}\,dt} \biggr) \biggr\vert \biggr] \\ &\quad =\frac{(b_{2}-b_{1})^{2}}{\alpha+1} \biggl[\frac{1}{2(\alpha+1)} \biggl\vert \psi'' \biggl(\frac{3\alpha b_{1}+2b_{1}+b_{2}}{3(\alpha+1)} \biggr) \biggr\vert + \frac{1}{\alpha+2} \biggl\vert \psi'' \biggl( \frac{\alpha b_{1}+2b_{1}+b_{2}}{\alpha+3} \biggr) \biggr\vert \biggr]. \end{aligned}$$

 □

Remark 2.12

In Theorem 2.11, if we take \(\alpha=1\), then we obtain

$$\begin{aligned} &\biggl\vert \psi \biggl(\frac{b_{1}+b_{2}}{2} \biggr)-\frac{1}{b_{2}-b_{1}} \int_{b_{1}}^{b_{2}}\psi(t)\,dt \biggr\vert \\ &\quad \leq \frac{(b_{2}-b_{1})^{2}}{2} \biggl[\frac{1}{4} \biggl\vert \psi'' \biggl(\frac{5b_{1}+b_{2}}{6} \biggr) \biggr\vert +\frac{1}{3} \biggl\vert \psi'' \biggl(\frac{3b_{1}+b_{2}}{4} \biggr) \biggr\vert \biggr]. \end{aligned}$$

Theorem 2.13

Assume that \(\psi \in C^{2}([b_{1},b_{2}])\) and \(|\psi''|\) is a concave function. Then, for any \(\alpha>0\), we have the inequality

$$\begin{aligned} & \biggl\vert \frac{\alpha\psi(b_{1})+\psi(b_{2})}{\alpha+1}-\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \biggr\vert \\ &\quad \leq\frac{\alpha(b_{2}-b_{1})^{2}}{2(\alpha+1)(\alpha+2)} \biggl\vert \psi'' \biggl( \frac{2\alpha b_{1}+\alpha b_{2}+4b_{1}+5b_{2}}{3(\alpha+3)} \biggr) \biggr\vert . \end{aligned}$$

Proof

From (2.21) one has

$$\begin{aligned} &\frac{\alpha\psi(b_{1})+\psi(b_{2})}{\alpha+1}-\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \\ &\quad =\frac{(b_{2}-b_{1})^{2}}{\alpha+1} \int_{0}^{1} \bigl(t-t^{\alpha+1} \bigr) \psi'' \bigl(tb_{1}+(1-t)b_{2} \bigr) \,dt. \end{aligned}$$

Taking absolute on both sides and using Jensen’s integral inequality, we get

$$\begin{aligned} & \biggl\vert \frac{\alpha\psi(b_{1})+\psi(b_{2})}{\alpha+1}-\frac{\Gamma(\alpha+1)}{(b_{2}-b_{1})^{\alpha}}J_{b_{1}+}^{\alpha} \psi(b_{2}) \biggr\vert \\ &\quad \leq\frac{(b_{2}-b_{1})^{2}}{\alpha+1} \int_{0}^{1} \bigl(t-t^{\alpha+1} \bigr)\,dt \biggl\vert \psi'' \biggl(\frac{\int_{0}^{1} (t-t^{\alpha+1} )\{tb_{1}+(1-t)b_{2}\}\,dt}{\int_{0}^{1} (t-t^{\alpha+1} )\,dt} \biggr) \biggr\vert \\ &\quad =\frac{\alpha(b_{2}-b_{1})^{2}}{2(\alpha+1)(\alpha+2)} \biggl\vert \psi'' \biggl( \frac{2\alpha b_{1}+\alpha b_{2}+4b_{1}+5b_{2}}{3(\alpha+3)} \biggr) \biggr\vert . \end{aligned}$$

 □

Remark 2.14

Let \(\alpha=1\). Then Theorem 2.13 leads to

$$ \biggl\vert \frac{\psi(b_{1})+\psi(b_{2})}{2}-\frac{1}{b_{2}-b_{1}} \int_{b_{1}}^{b_{2}}\psi(t)\,dt \biggr\vert \leq \frac{(b_{2}-b_{1})^{2}}{12} \biggl\vert \psi'' \biggl( \frac{b_{1}+b_{2}}{2} \biggr) \biggr\vert . $$

Remark 2.15

We can also get the same results as in this article if we use the Green’s function \(G_{2}\) given by (2.2) instead of the Green’s function \(G_{1}\) given by (2.1). But we can only get previously known results if we use the idea of the article and the Green’s function \(G_{3}\) given by (2.3) or \(G_{4}\) given by (2.4). This is the reason why we only deal with the Green’s function \(G_{1}\) in the article. As exercises, interested readers can use other three Green’s functions to give their corresponding results.

Results and discussion

In the article, we use Green’s function method to establish the left Riemann–Liouville fractional Hermite–Hadamard type inequalities. The given idea by using the Green’s function (2.1) or any other new Green’s function may be furthered to research the Hermite–Hadamard inequality for fractional integrals as presented in [38] and to research the Hermite–Hadamard inequality for pre-invex, s-convex, co-ordinate convex functions etc.

Conclusion

In the article, we establish the left Riemann–Liouville fractional Hermite–Hadamard type inequalities and the generalized Hermite–Hadamard type inequalities by using Green’s function and Jensen’s inequality. The given idea and results are novel and interesting, they may stimulate further research in the theory of fractional integrals and generalized convex functions.

References

  1. Roberts, A.W., Varberg, D.E.: Convex Functions. Academic Press, New York (1973)

    MATH  Google Scholar 

  2. Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability. Springer, Berlin (1989)

    Book  MATH  Google Scholar 

  3. Pečarić, J.E., Proschan, F., Tong, Y.L.: Convex Functions, Partial Orderings, and Statistical Applications. Academic Press, Boston (1992)

    MATH  Google Scholar 

  4. Udrişte, C.N.: Convex Functions and Optimization Methods on Riemannian Manifolds. Kluwer Academic Publishers Group, Dordrecht (1994)

    Book  MATH  Google Scholar 

  5. Pearce, C.E.M., Pečarić, J.E.: An integral inequality for convex functions, with applications to teletraffic congestion problems. Math. Oper. Res. 20(3), 526–528 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Dedić, L., Pearce, C.E.M., Pečarić, J.E.: The Euler formulae and convex functions. Math. Inequal. Appl. 3(2), 211–221 (2000)

    MathSciNet  MATH  Google Scholar 

  7. Niculescu, C.P., Persson, L.-E.: Convex Functions and Their Applications. Springer, New York (2006)

    Book  MATH  Google Scholar 

  8. Breckner, W.W., Trif, T.: Convex Functions and Related Functional Equations. Presa Universitară Clujeană, Cluj-Napoca (2008)

    MATH  Google Scholar 

  9. Borwein, J.M., Vanderwerff, J.D.: Convex Functions: Constructions, Characterizations and Counterexamples. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  10. Wang, M.-K., Chu, Y.-M.: Landen inequalities for a class of hypergeometric functions with applications. Math. Inequal. Appl. 21(2), 521–537 (2018)

    MathSciNet  MATH  Google Scholar 

  11. Wang, M.-K., Qiu, S.-L., Chu, Y.-M.: Infinite series formula for Hübner upper bounds function with applications to Hersch–Pfluger distortion function. Math. Inequal. Appl. 21(3), 629–648 (2018)

    Google Scholar 

  12. Adil Khan, M., Chu, Y.-M., Kashuri, A., Liko, R.: Hermite–Hadamard type fractional integral inequalities for \(\mathrm{MT}_{(r;g,m,\varphi)}\)-preinvex functions. J. Comput. Anal. Appl. 26(8), 1487–1503 (2019)

    Google Scholar 

  13. Pearce, C.E.M., Pečarić, J.E.: An inequality for convex functions. J. Math. Anal. Appl. 183(3), 523–527 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  14. Tseng, K.-L., Hwang, S.-R., Dragomir, S.S.: New Hermite–Hadamard-type inequalities for convex function II. Comput. Math. Appl. 62(1), 401–418 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dragomir, S.S.: Bounds for convex functions of Čebyšev functional via Sonin’s identity with applications. Commun. Math. 22(2), 107–132 (2014)

    MathSciNet  MATH  Google Scholar 

  16. Pečarić, J.E., Smoljak, K.K.: New Steffensen type inequalities involving convex functions. Results Math. 67(1–2), 217–234 (2015)

    MathSciNet  MATH  Google Scholar 

  17. Pavić, Z.: Certain inequalities for convex functions. J. Math. Inequal. 9(4), 1349–1364 (2015)

    MathSciNet  MATH  Google Scholar 

  18. Özdemir, M.E., Ekinci, A.: Generalized integral inequalities for convex functions. Math. Inequal. Appl. 19(4), 1429–1439 (2016)

    MathSciNet  MATH  Google Scholar 

  19. Liu, W.-J., Zhuang, H.-F.: Some quantum estimates of Hermite–Hadamard inequalities for convex functions. J. Appl. Anal. Comput. 7(2), 501–522 (2017)

    MathSciNet  Google Scholar 

  20. Yang, Z.-H., Zhang, W., Chu, Y.-M.: Sharp Gautschi inequality for parameter \(0< p<1\) with applications. Math. Inequal. Appl. 20(4), 1107–1120 (2017)

    MathSciNet  MATH  Google Scholar 

  21. Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On approximating the error function. Math. Inequal. Appl. 21(2), 469–479 (2018)

    MathSciNet  MATH  Google Scholar 

  22. Yang, Z.-H., Qian, W.-M., Chu, Y.-M., Zhang, W.: On approximating the arithmetic-geometric mean and complete elliptic integral of the first kind. J. Math. Anal. Appl. 462(2), 1714–1726 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  23. Noor, M.A.: Hermite–Hadamard integral inequalities for log-φ-convex functions. Nonlinear Anal. Forum 13(1), 119–124 (2008)

    MathSciNet  MATH  Google Scholar 

  24. Kikianty, E.: Hermite–Hadamard inequality in the geometry of banach spaces. Ph.D. thesis, Victoria University (2010) Available online at http://vuir.vu.edu.au/15793/1/EderKikiantyThesis.pdf

  25. Niculescu, C.P., Persson, L.-E.: Old and new on the Hermite–Hadamard inequality. Real Anal. Exch. 29(2), 663–685 (2003/04)

  26. Dragomir, S.S., Agarwal, R.P.: Two new mappings associated with Hadamard’s inequalities for convex functions. Appl. Math. Lett. 11(3), 33–38 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  27. Dragomir, S.S., Agarwal, R.P.: 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)

    Article  MathSciNet  MATH  Google Scholar 

  28. Pearce, C.E.M., Pečarić, J.E.: Inequalities for differentiable mappings with applications to special means and quadrature formulae. Appl. Math. Lett. 13(2), 51–55 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kirmaci, U.S., Klaričić Bakula, M., Özdemir, M.E., Pečarić, J.E.: Hadamard-type inequalities for s-convex functions. Appl. Math. Comput. 193(1), 26–35 (2007)

    MathSciNet  MATH  Google Scholar 

  30. Özdemir, M.E., Avcı, M., Set, E.: On some inequalities of Hermite–Hadamard type via m-convexity. Appl. Math. Lett. 23(9), 1065–1070 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Set, E., Karataş, S.S., Adil Khan, M.: Hermite–Hadamard type inequalities obtained via fractional integrals for differentiable m-convex and \((\alpha, m)\)-convex functions. Int. J. Anal. 2016, Article ID 4765691 (2016)

    MathSciNet  MATH  Google Scholar 

  32. Adil Khan, M., Khurshid, Y., Ali, T., Rehman, N.: Inequalities for three times differentiable functions. Punjab Univ. J. Math. 48(2), 35–48 (2016)

    MathSciNet  MATH  Google Scholar 

  33. Wang, M.-K., Li, Y.-M., Chu, Y.-M.: Inequalities and infinite product formula for Ramanujan generalized modular equation function. Ramanujan J. 46(1), 189–200 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  34. 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)

    MathSciNet  MATH  Google Scholar 

  35. Chu, Y.-M., Adil Khan, M., Ali, T., Dragomir, S.S.: Inequalities for α-fractional differentiable functions. J. Inequal. Appl. 2017, Article ID 93 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  36. Chu, Y.-M., Adil Khan, M., Khan, T.U., Khan, J.: Some new inequalities of Hermite–Hadamard type for s-convex functions with applications. Open Math. 15, 1414–1430 (2017)

    MathSciNet  MATH  Google Scholar 

  37. Adil Khan, M., Begum, S., Khurshid, Y., Chu, Y.-M.: Ostrowski type inequalities involving conformable fractional integrals. J. Inequal. Appl. 2018, Article ID 70 (2018)

    Article  MathSciNet  Google Scholar 

  38. Sarikaya, M.Z., Set, E., Yaldiz, H., Basak, N.: Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities. Math. Comput. Model. 57(9), 2403–2407 (2013)

    Article  MATH  Google Scholar 

  39. Kunt, M., Karapınar, D., Turhan, S., İşcan, İ.: The left Rieaman–Liouville fractional Hermite–Hadamard type inequalities for convex functions. RGMIA Res. Rep. Collect. 20, Article ID 101 (2017)

    Google Scholar 

  40. Mehmood, N., Agarwal, R.P., Butt, S.I., Pečarić, J.E.: New generalizations of Popoviciu-type inequalities via new Green’s functions and Montgomery identity. J. Inequal. Appl. 2017, Article ID 108 (2017)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

The research was supported by the Natural Science Foundation of China (Grants Nos. 61673169, 11601485, 11701176) and the Natural Science Foundation of the Department of Education of Zhejiang Province (Grant No. Y201635325).

Author information

Authors and Affiliations

Authors

Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Yu-Ming Chu.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Adil Khan, M., Iqbal, A., Suleman, M. et al. Hermite–Hadamard type inequalities for fractional integrals via Green’s function. J Inequal Appl 2018, 161 (2018). https://doi.org/10.1186/s13660-018-1751-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/s13660-018-1751-6

MSC

  • 26D15

Keywords

  • Hermite–Hadamard inequality
  • convex function
  • Green’s function