On Caputo fractional derivative inequalities by using strongly $(\alpha, h-m)$-convexity

1.   Introduction

The Hadamard inequality was established by Charles Hermite (1883) and Jacques Hadamard (1893); they discovered it independently, see ^[1,2]. In recent years along with other celebrated mathematical inequalities, various authors have studied the Hadamard inequality very frequently. Especially, it is analyzed for fractional integral and derivative operators at large number, see ^{[3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]} and references therein.

A very useful notion of convex function was defined at the start of twentieth century due to the Hadamard inequality, it provides necessary and sufficient conditions for a function to be convex. A lot of variants of this inequality have been established by applying functions directly linked with convex function. The mathematical inequality (1.1) satisfied by convex functions, motivates to give new definitions which broaden the notion of convexity elegantly. For instance the notions of $m$-convex, $h$-convex, $(\alpha, m)$-convex, $(s, m)$-convex, $(h-m)$-convex, $p$-convex, $(p, h)$-convex and strongly convex functions have been established in literature by modifying the inequality (1.1) conveniently. The variants of Hadamard inequality for these functions provide generalizations and refinements of the classical variant.

The main goal of this paper is to study the Caputo fractional derivative versions of the Hadamard inequality for strongly $(\alpha, h-m)$-convex functions which provide at the same time refinements as well as generalizations of various well-known inequalities exist in literature. The error bounds of some Caputo fractional derivative inequalities by applying well-known identities are given. Next, we give preliminary definitions which are useful for establishing the results of this paper.

Definition 1.1. ^[19] A function $\zeta:J\rightarrow \mathbb{R}$ is called convex if it satisfies the inequality:

where $u\in[0, 1]$ and $x, y\in J, J$ is an interval in $\mathbb{R}$.

If (1.1) holds in the reverse order, then $\zeta$ is called concave function.

Convex functions can be visualized elegantly by the Hadamard inequality. It is stated as follows:

Theorem 1.2. ^[20] Suppose that $\zeta:J \rightarrow \mathbb{R}$ is a convex function. For $a, b\in J$ with $a < b$, we have the following inequality

If orders in (1.2) are reversed, then $\zeta$ must be concave function.

Convex functions have been extended in many ways to obtain the generalizations and refinements of well-known inequalities. For example, operator $m$-convex functions are defined in ^[21], $m$-convex functions on set valued functions are defined in ^[22]. A recently introduced definition of $(\alpha, h-m)$ convex function unifies the well-known classes of $(s, m)$-convex, $(h-m)$-convex and $(\alpha, m)$-convex functions along with many other kinds of convexities.

Definition 1.3. ^[23] Let $h:J\to\mathbb{R}$ be a non-negative function, where $J\subseteq\mathbb{R}$ is an interval that contains $(0, 1)$. A function $\zeta:[0, b]\to\mathbb{R}$ is called $(\alpha, h-m)$-convex function, if $\zeta$ is non-negative and satisfies the inequality:

for all $x, y\in[0, b],$ $u\in(0, 1)$ and $(\alpha, m)\in[0, 1]^2$.

Remark 1.4. The inequality (1.3) provides the definitions of $(\alpha, m)$ convex function for $h(u) = u$; $(h-m)$-convex function for $\alpha = 1; (s, m)$-convex function for $h(u) = u^s, \alpha = 1; m$-convex function for $h(u) = u, \alpha = 1; s$-convex function for $m = 1$, $\alpha = 1$, $h(u) = u^s$; $h$-convex function for $m = 1, \alpha = 1$; convex function for $h(u) = u, \alpha = 1, m = 1$; Godunova-Levin function for $h(u) = \frac{1}{u}, \alpha = 1, m = 1$ and $s$-Godunova-Levin function for $h(u) = \frac{1}{u^s}, \alpha = 1, m = 1$.

The definition of strongly convex function was introduced by Polyak as follows:

Definition 1.5. ^[24] Let $(\mathbb{X}, \|\cdot\|)$ be a normed space. A function $\zeta:D\subset \mathbb{X}\rightarrow \mathbb{R}$ is called strongly convex function with modulus $C\geq0$, if it satisfies the inequality:

for all $x, y\in {D}$, where $D$ is a convex subset of $\mathbb{X}$ and $u\in[0, 1]$.

For further properties, applications and utilization of strongly convex functions we refer the readers to ^{[25,26,27,28,29,30,31,32]}. We give the definition of strongly $(\alpha, h-m)$-convex function as follows:

Definition 1.6. ^[33] Let $h:J\to\mathbb{R}$ be a non-negative function, where $J\subseteq\mathbb{R}$ is an interval which contains $(0, 1)$. A function $\zeta:[0, b]\to\mathbb{R}$ is called strongly $(\alpha, h-m)$-convex function with modulus $C\geq0$, if $\zeta$ is non-negative and satisfies the inequality:

for $(\alpha, m)\in[0, 1]^2, x, y\in[0, b]$ and $u\in(0, 1)$.

The inequality (1.5) provides the definitions of strongly convex and convex, strongly $h$-convex and $h$-convex, strongly $m$-convex and $m$-convex, strongly $(s, m)$-convex and $(s, m)$-convex, strongly $(\alpha, m)$-convex and $(\alpha, m)$-convex, strongly $(h-m)$-convex and $(h-m)$-convex functions in specific cases.

Next, we give definition of the CFD.

Definition 1.7. ^[34] Let $\zeta \in AC^n[a, b]$ and $n = [\Re(\gamma)]+1$. Then CFD of order $\gamma\in\mathbb{C}, \Re(\gamma) > 0$ of the function $\zeta$ are defined by:

and

If $\gamma = n \in \{1, 2, 3, \ldots\}$ and usual derivative of order $n$ exists, then CFD $\left({^C}D^{\gamma}_{a+}\zeta\right)(x)$ coincides with $\zeta^{(n)}(x)$, whereas $\left({^C}D^{\gamma}_{b-}\zeta\right)(x)$ coincides with $\zeta^{(n)}(x)$ with exactness to a constant multiplier $(-1)^n$.

In particular we have

where $n = 1$ and $\gamma = 0$.

For establishing the Hadamard type inequalities, the definitions of Caputo derivative and strongly $(\alpha, h-m)$-convex function are the key factors. To study the error estimates of Caputo fractional Hadamard type inequalities the following two lemmas are very useful. We also utilize the well-known integral versions of the Hölder and power mean inequalities.

Lemma 1.8. ^[8] Let $0\leq a < b$ and $\zeta\in C^{n+1}[a, b]$. Then one can have the following identity for CFD

Lemma 1.9. ^[9] Let $0\leq a < mb, \, m\in(0, 1]$ and $\zeta\in C^{n+1}[a, b]$. Then one can have the following identity for CFD

with ${\gamma} > 0$.

The rest of paper is composed of the following sections: In Section 2, we give refinements of two Hadamard type inequalities for strongly $(\alpha, h-m)$-convex functions via CFD. It is noted that the particular cases of the main results are connected with classical results already published in the near past publish articles. In Section 3, by applying the identities (1.9) and (1.10), we find improvements of error bounds of various inequalities.

2.   Main results

The following result provides Hadamard inequality for strongly $(\alpha, h-m)$-convex function for CFD. It will reproduce many CFD inequalities for various kinds of convex functions.

Theorem 2.1. Suppose that $\zeta\in C^{n}[a, b]$ and $\zeta^{(n)}$ satisfies the inequality (1.5). Then it must satisfy the CFD inequality:

where $m\in (0, 1], \, 0\leq{a} < mb$ and ${\gamma} > 0$.

Proof. As $\zeta^{(n)}$ is strongly $(\alpha, h-m)$-convex function, from (1.5) one can easily have

Let $x = (1-u)\frac{a}{m}+ub\leq b$ and $y = m(1-u)b+ua\geq a$, where $u\in[0, 1]$. Then one can have

By multiplying (2.3) with $u^{n-\gamma-1}$ and integrating over $[0, 1]$, we obtain

By applying Definition 1.7 and after a little computation, we have

As $\zeta^{(n)}$ is strongly $(\alpha, h-m)$-convex function, from (1.5) one can easily have

By multiplying (2.6) with $h\left(1-\frac{1}{2^\alpha}\right)u^{n-\gamma-1}$ and integrating over $[0, 1]$ we obtain

By using Definition 1.7, one can have

Again by using strongly $(\alpha, h-m)$-convexity for the function $\zeta^{(n)}$ and adopting the same pattern as we did for inequality (2.6), we have the following inequality

By adding (2.8) and (2.9), we have

Inequalities (2.5) and (2.10) constituted the required inequality.

The following result gives another variant of Hadamard inequality for strongly $(\alpha, h-m)$-convex function for CFD. It will reproduce many CFD inequalities for various kinds of convex functions.

Theorem 2.2. Under the supposition of Theorem 2.1, the following inequality holds:

with ${\gamma} > 0$.

Proof. Let $x = \frac{a}{m}\left(\frac{2-u}{2}\right)+b\frac{u}{2}$ and $y = a\frac{u}{2}+m\left(\frac{2-u}{2}\right)b$, where $u\in[0, 1]$ in (2.2). Then one can have the inequality in the following form

By multiplying (2.12) with $u^{n-\gamma-1}$ and then integrating over $[0, 1],$ one can have

By using Definition 1.7, we have

As $\zeta^{(n)}$ is strongly $(\alpha, h-m)$-convex function, from (1.5) one can easily have

By multiplying (2.15) with $h\left(1-\frac{1}{2^\alpha}\right)u^{n-\gamma-1}$ and integrating over $[0, 1]$, one can have

With the help of Definition 1.7, one can see that

Also, using the definition of strongly $(\alpha, h-m)$-convex function $\zeta^{(n)}$ and working on the same lines as we did for the inequality (2.15), one can have the following inequality

By adding (2.17) and (2.18), we get

From (2.14) and (2.19), (2.11) can be obtained.

Theorems 2.1 and 2.2 are connected with many results, which are explained in the following remark.

Remark 2.3. The inequalities (2.1) and (2.11) provide the fractional Hadamard inequalities for strongly $(\alpha, m)$ convex function for $h(u) = u$; strongly $(h-m)$-convex function for $\alpha = 1$; strongly $(s, m)$-convex function for $h(u) = u^s, \alpha = 1$; strongly $m$-convex function for $h(u) = u, \alpha = 1$; strongly convex function for $h(u) = u, \alpha = 1, m = 1$. For $C = 0$ the inequalities (2.1) and (2.11) give the Hadamard inequalities for $(\alpha, h-m)$-convex function which further gives the Hadamard inequality for: Convex, $s$-convex, $h$-convex, $m$-convex, $(\alpha, m)$-convex, $(h-m)$-convex and $(s, m)$-convex functions respectively.

3.   More fractional Hadamard type inequalities for CFD

Here we give Hadamard type inequalities by using identities for CFD, which are stated in Lemmas 1.8 and 1.9.

Theorem 3.1. Suppose that $\zeta\in C^{n+1}[a, b]$ and $|\zeta^{(n+1)}|$ satisfies the inequality (1.5) and $h(x+y)\leq h(x)h(y)$. Then it satisfies the CFD inequality:

with $0\leq{a} < mb, {\gamma} > 0$ and $\frac{1}{p}+\frac{1}{q} = 1.$

Proof. As $|\zeta^{(n+1)}|$ is strongly $(\alpha, h-m)$-convex function, one can have

From the definition of strongly $(\alpha, h-m)$-convexity of $|\zeta^{(n+1)}|$ and Lemma 1.8, one can have

Now, by applying Hölder's inequality on the right side of inequality (3.2), we have

and

By using values of integrals from (3.3) and (3.4) in (3.2), we obtain the inequality (3.1).

Corollary 3.2. The class of $(\alpha, m)$-convex functions satisfies the CFD inequality as follows:

Proof. In the inequality (3.1) of Theorem 3.1 if one set $C = 0$ and $h(u) = u$, we obtain the inequality (3.5).

Corollary 3.3. The class of $m$-convex functions satisfies CFD inequality as follows:

Proof. In the inequality (3.1) of Theorem 3.1, if one set $C = 0, \alpha = 1$ and $h(u) = u$, we obtain the inequality (3.6).

In the following theorem, we give an error bound of inequality (2.11) with the help of Lemma 1.9.

Theorem 3.4. Suppose that $\zeta\in C^{n+1}[a, b]$ and $|\zeta^{(n+1)}|^q$ with $q > 1$ satisfies the inequality (1.4). Then it satisfies the CFD inequality:

with $0\leq{a} < mb$.

Proof. By using Lemma 1.9 for $k = 1$. Then applying modulus property and using power mean inequality and the strongly $(\alpha, h-m)$-convexity of $|\zeta^{(n+1)}|^q$, we have

The proof is completed.

Theorem 3.5. Under the assumptions of Theorem 3.4, the following inequality holds:

Proof. By using Lemma 1.9 for $k = 1$. Then applying modulus inequality and Holder's inequality, we have

By applying Holder's inequality, we have

By applying strongly $(\alpha, h-m)$-convexity of $|\zeta^{(n+1)}|^q$, we have

The proof is completed.

Remark 3.6. The inequalities (3.7) and (3.8) provide the fractional Hadamard inequalities for strongly $(\alpha, m)$ convex function for $h(u) = u$; strongly $(h-m)$-convex function for $\alpha = 1$; strongly $(s, m)$-convex function for $h(u) = u^s$, $\alpha = 1$; strongly $m$-convex function for $h(u) = u$, $\alpha = 1$; strongly convex function for $h(u) = u$, $\alpha = 1$, $m = 1$. For $C = 0$ the inequalities (3.7) and (3.8) give the Hadamard inequalities for $(\alpha, h-m)$-convex function which further give the Hadamard inequalities for $(h-m)$-convex, $(\alpha, m)$-convex, $(s, m)$-convex, $m$-convex, $h$-convex, $s$-convex and convex functions respectively.

4.   Conclusions

In this article CFD inequalities of Hadamard type are combined in the form of compact results. These results generate many already published inequalities by selecting specific functions and parameters that appear in the inequality (1.5). Refinements of many Hadamard type inequalities for Caputo fractional derivatives are deducible. The connections of established results with known published results are shown in form of remarks and corollaries.

Acknowledgments

This research has received funding support from the National Science, Research and Innovation Fund (NSRF), Thailand.

Conflict of interest

The authors declare no conflict of interest.