Hadamard Inequalities for Strongly 
                     
                        
                           α
                           ,
                           m
                        
                     
                  -Convex Functions via Caputo Fractional Derivatives

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


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.
In the following, we give the definition of Caputo fractional derivatives.
en, Caputo fractional derivatives of order β ∈ C, R(β) > 0, of ψ are defined as follows: If β � n ∈ 1, 2, 3, . . . { } and usual derivative of order n exists, then Caputo fractional derivative ( C D β a + ψ )( x ) coincides with ψ (n) (x), where as ( C D β b − ψ )( x ) coincides with ψ (n) (x) with exactness to a constant multiplier (− 1) n . In particular, we have where n � 1 and β � 0. 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 D be a convex subset of L and (L, ‖ · ‖) be a normed space. A function ψ: D ⊂ L ⟶ R is called strongly convex function with modulus C if it satisfies For C � 0, (3) gives the inequality satisfied by convex functions. Many authors have been inventing the properties and applications of strongly convex functions, see [17][18][19][20][21]. e concept of m-convex functions and strongly m-convex functions are introduced in [22,23], respectively. In [23], the definition of m-convex functions is given.
In [22], strongly m-convex function is introduced as follows.
A well-known inequality named Hadamard inequality is another interpretation of convex function. It is stated as follows [31]. Theorem 1. Let ψ: I ⟶ R be a convex function on interval I ⊂ R and a, b ∈ I, where a < b. en, 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. ese 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.
e following identity is established in [34].
with β > 0. e 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. e aim of this paper is to prove the Hadamard inequality for Caputo fractional derivatives of strongly (α, m)-convex functions. We have obtained refinements of various inequalities proved for convex and m-convex functions.
In Section 2, we will give two versions of the Hadamard inequality for Caputo fractional derivatives using strongly (α, m)-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 wellknown inequalities.

Main Results
e following results give the Hadamard inequality for Caputo fractional derivatives of strongly (α, m)-convex functions.

Journal of Mathematics
Multiplying (13) with z n− β− 1 on both sides and making integration over [0, 1], we get By using change of the variables and computing the last integral, from (14), we get Further, it takes the following form: Since ψ (n) is strongly (α, m)-convex function with modulus C, for z ∈ [0, 1], then one has By multiplying (17) with (1 − (1/2 α ))z n− β− 1 on both sides and making integration over [0, 1], we get

Journal of Mathematics
By using change of the variables and computing the last integral, from (18), we get Further, it takes the following form: Since ψ (n) is strongly (α, m)-convex function with modulus C, for z ∈ [0, 1], then one has By multiplying (21) with (1/2 α )z n− β− 1 on both sides and making integration over [0, 1], we get By using change of the variables and computing the last integral, from (22), we get Further, it takes the following form: By adding (20) and (24), we have

Journal of Mathematics
By using change of variables and computing the last integral, from (36), we get Further, it takes the following form: By adding (34) and (38), we get From (30) and (39), (26) can be obtained.

Error Bounds of Fractional Hadamard Inequalities
In this section, we give refinements of the error bounds of fractional Hadamard inequalities for Caputo fractional derivatives.

Data Availability
No data were used in this paper.