Abstract

In this paper, we present two versions of the Hadamard inequality for convex functions via Caputo fractional derivatives. Several related results are analyzed for convex and -convex functions along with their refinements and generalizations. The error bounds of the Hadamard inequalities are established by applying some known identities.

1. Introduction

Fractional calculus is a natural extension of classical calculus and the notions related to integer order derivatives and integrals have been extended to fractional order derivatives and integrals. Many classical inequalities related to integrals of real valued functions have been presented for fractional integrals. The Hadamard inequality is one of them which is studied extensively for different types of convex functions via fractional derivatives and integrals. For the detailed study of Hadamard fractional integral inequalities, we refer the readers to [1–13]. In 1967, Caputo made the most significant contributions to fractional calculus by reformulating the definition of the Riemann–Liouville fractional derivatives [14].

In the following, we give the definition of Caputo fractional derivatives.

Definition 1 [15]. Let and . Then, Caputo fractional derivatives of order , , of are defined as follows:If and usual derivative of order exists, then Caputo fractional derivative coincides with , where as coincides with with exactness to a constant multiplier . In particular, we havewhere and .
Convex functions are represented in terms of different inequalities. Many of the well-known inequalities are consequences of convex functions. Strongly convexity is a strengthening of the notion of convexity; some properties of strongly convex functions are just “stronger versions” of known properties of convex functions. Strongly convex function was introduced by Polyak [16].

Definition 2. Let be a convex subset of and be a normed space. A function is called strongly convex function with modulus if it satisfiesFor , (3) gives the inequality satisfied by convex functions. Many authors have been inventing the properties and applications of strongly convex functions, see [17–21].
The concept of -convex functions and strongly -convex functions are introduced in [22, 23], respectively. In [23], the definition of -convex functions is given.

Definition 3. The function is said to be -convex, where , if for every and we haveIn [22], strongly -convex function is introduced as follows.

Definition 4. A function is called strongly -convex function with modulus ifwhere and .
In [24], -convex function is introduced as follows.

Definition 5. A function is said to be -convex function, where and , if for every and , we haveFrom -convex functions, one can obtain star-shaped, -convex, convex, and -convex functions. In literature, -convex functions are considered by many researchers and mathematicians and many properties especially inequalities have been obtained for them, for example (see [25–30]).
The strongly -convex function is introduced as follows.

Definition 6. A function is said to be strongly -convex function with modulus , for , ifholds for all and .
A well-known inequality named Hadamard inequality is another interpretation of convex function. It is stated as follows [31].

Theorem 1. Let be a convex function on interval and , where . Then, the following inequality holds:

If order in (8) is reversed, then it holds for concave function.

Fractional integral inequalities are useful in establishing the uniqueness of solutions for certain fractional partial differential equations. These inequalities also provide upper as well as lower bounds for solutions of the fractional boundary value problems. Fractional integral inequalities are in the study of several mathematicians. For fractional versions of Hadamard inequality, we refer the researchers and references [1–5, 11, 12, 32].

Farid et al. [33] established the following identity for Caputo fractional derivatives.

Lemma 1. Let , , be the function such that . Also, let be positive and convex function on . Then, the following equality for Caputo fractional derivatives holds:

The following identity is established in [34].

Lemma 2. Let be a differentiable mapping on with . If , then the following equality for Caputo fractional derivatives holds:

with .

The Hadamard inequality for Caputo fractional derivatives of convex functions is studied in [7, 33, 34]; also, the error estimations are established by using aforementioned identities. The aim of this paper is to prove the Hadamard inequality for Caputo fractional derivatives of strongly -convex functions. We have obtained refinements of various inequalities proved for convex and -convex functions.

In Section 2, we will give two versions of the Hadamard inequality for Caputo fractional derivatives using strongly -convex functions. Also, we connect the particular cases with some classical results. In Section 3, by applying known identities, we will derive refinements of some well-known inequalities.

2. Main Results

The following results give the Hadamard inequality for Caputo fractional derivatives of strongly -convex functions.

Theorem 2. Let be a positive function with , , and . If is a strongly -convex function with modulus , then the following inequality for Caputo fractional derivatives holds:with and .

Proof. Since is strongly -convex function with modulus , for , we haveLet and , . Then, we haveMultiplying (13) with on both sides and making integration over , we getBy using change of the variables and computing the last integral, from (14), we getFurther, it takes the following form:Since is strongly -convex function with modulus , for , then one hasBy multiplying (17) with on both sides and making integration over , we getBy using change of the variables and computing the last integral, from (18), we getFurther, it takes the following form:Since is strongly -convex function with modulus , for , then one hasBy multiplying (21) with on both sides and making integration over , we getBy using change of the variables and computing the last integral, from (22), we getFurther, it takes the following form:By adding (20) and (24), we haveInequalities (16) and (25) constituted the required inequality.
The consequences of Theorem 2 are stated in the following remark.