Generalized Hadamard Fractional Integral Inequalities for Strongly (s, m)-Convex Functions

Fractional calculus is related to the integrals and derivatives of any arbitrary real or complex order. Its history starts from the end of the seventeenth century, but now it has many applications in almost every field of mathematics, science, and engineering such as electromagnetic, viscoelasticity, fluid mechanics, and signal processing. Fractional integral and derivative operators are of great importance in fractional calculus. ,e Riemann–Liouville fractional integrals are playing key role in its development. Sarikaya et al. [1, 2] studied Hadamard inequality through Riemann–Liouville fractional integrals of convex functions. ,is study has encouraged a number of researchers to work further in the field of mathematical inequalities by using fractional integral operators. As a consequence, Hadamard’s inequality is generalized and extended by fractional integral operators in many ways (see [3–9] and the references therein). ,e following inequality is the well-known Hadamard inequality for convex functions which is stated in [10]. Let f: I⟶ R be a convex function defined on an interval I ⊂ R and x, y ∈ I where x<y. ,en, the following inequality holds:


Introduction
Fractional calculus is related to the integrals and derivatives of any arbitrary real or complex order. Its history starts from the end of the seventeenth century, but now it has many applications in almost every field of mathematics, science, and engineering such as electromagnetic, viscoelasticity, fluid mechanics, and signal processing. Fractional integral and derivative operators are of great importance in fractional calculus. e Riemann-Liouville fractional integrals are playing key role in its development. Sarikaya et al. [1,2] studied Hadamard inequality through Riemann-Liouville fractional integrals of convex functions.
is study has encouraged a number of researchers to work further in the field of mathematical inequalities by using fractional integral operators. As a consequence, Hadamard's inequality is generalized and extended by fractional integral operators in many ways (see [3][4][5][6][7][8][9] and the references therein). e following inequality is the well-known Hadamard inequality for convex functions which is stated in [10].
Let f: I ⟶ R be a convex function defined on an interval I ⊂ R and x, y ∈ I where x < y. en, the following inequality holds: For the history of this inequality, we refer the readers to [11,12]. Use of convex functions in the fields of statistics [13], economics [14], and optimization [15] is of prime importance because they play an important role in development of new concepts and notions. Various scholars extended the research on integral inequalities to fractal sets [16]. In this paper, the Hadamard inequality is studied for generalized Riemann-Liouville fractional integrals of strongly (s, m)-convex functions; also, by using two integral identities, some error bounds of already established fractional inequalities are studied. Bracamonte et al. [17] defined the strongly (s, m)-convex function as follows.
By establishing an integral identity, the following error estimation of inequality (6) is proved.
Theorem 3 (see [1]). Let f: , then the following fractional integral inequality holds: A k-analogue of Riemann-Liouville integral is defined as follows.
Theorem 4 (see [22]). Let f: , then the following inequality for k-fractional integrals holds: Theorem 5 (see [23]). Under the assumption of eorem 4, the following inequality for k-fractional integrals holds: By establishing an integral identity, in the following theorem, the error estimation of eorem 4 is proved.
Theorem 6 (see [22]). Let f: , then the following inequality for k-fractional integrals holds: In the following, we recall the definition of generalized Riemann-Liouville fractional integrals by a monotonically increasing function.
Definition 4 (see [24]). Let f: [a, b] ⟶ R be an integrable function. Also, let ψ be an increasing and positive function on If ψ is identity function, then (14) and (15) coincide with (3) and (4). e k-analogue of generalized Riemann-Liouville fractional integral is defined as follows.
For further study of fractional integrals, see [26,27]. We will utilize the following well-known hypergeometric, beta, and incomplete beta functions in our results [28].
Journal of Mathematics 3 e rest of the paper is organized as follows. In Section 2, we obtain Hadamard inequalities for generalized Riemann-Liouville fractional integrals of strongly (s, m)-convex functions. Many specific cases are given as outcomes of these inequalities; they are related to the results which have been published in different papers. In Section 3, by using two integral identities for generalized fractional integrals, the error bounds of fractional Hadamard inequalities are established for differentiable strongly (s, m)-convex functions. is paper reproduces the results which are explicitly given in [1,2,22,23,[29][30][31][32][33][34][35][36][37]].

Main Results
is section is dedicated to the Hadamard inequality for strongly (s, m)-convex functions via generalized Riemann-Liouville fractional integrals. We will give two versions of this inequality. First one is stated and proved in the following theorem.
Proof. e following inequality holds for strongly (s, m)-convex functions.
By setting (20), multiplying resulting inequality with t (μ/k)− 1 , and then integrating with respect to t, we get (21) and by applying Definition 5, we get the following inequality: e above inequality leads to the first inequality of (19). On the other hand, f is strongly (s, m)-convex function with modulus c; for t ∈ [0, 1], we have the following inequality: By integrating (23) over [0, 1] after multiplying with t (μ/k)− 1 , the following inequality holds: Again using substitutions as considered in (21), we get is leads to the second inequality of (19). in (19), then eorem 4 is obtained. (iii) If c � 0, s � 1, m � 1, k � 1, and ψ is the identity function in (19), then eorem 1 is obtained. (iv) If k � 1, s � 1, m � 1, and ψ is the identity function in (19), then refinement of eorem 1 is obtained.

Error Estimations of Hadamard Inequalities via Strongly (s, m)-Convex Functions
In this section, we will study error estimations of Hadamard inequalities for generalized Riemann-Liouville fractional integrals of strongly (s, m)-convex functions. e estimations obtained here provide refinements of many wellknown results. e Mathematica program is used for integration. We recall the well-known Hölder's integral inequality.
with equality holding iff A|f(x)| p � B|g(x)| q almost everywhere, where A and B are constants.
In order to prove the next result, the following lemma is useful.