Inequalities for Riemann–Liouville Fractional Integrals of Strongly ( s, m ) -Convex Functions

The results of this paper provide two Hadamard-type inequalities for strongly ( s,m ) -convex functions via Riemann–Liouville fractional integrals and error estimations of well-known fractional Hadamard inequalities. Their special cases are given and connected with the results of some published papers.


Introduction
e most prominent inequality for convex functions is the well-known Hadamard inequality stated in the following.
Theorem 1 (see [1]). Let f: I ⊂ R ⟶ R be a convex function on the interval I, where x, y ∈ I with x < y. en, the following inequality holds: Convex functions are extended, generalized, and refined in different ways to define new types of convex functions. For instance, s-convex, m-convex, (s, m)-convex, strongly convex, and strongly (s, m)-convex functions are extensions of convex functions. e aim of this paper is to establish integral inequalities by using the class of strongly (s, m)-convex functions. We give definitions of (s, m)-convex and strongly (s, m)-convex functions as follows.
Next we give definition of Riemann-Liouville fractional integrals J α x + f and J α y − f which are utilized to get the desired results of this paper.
en, Riemann-Liouville fractional integral operators of order α > 0 are given by e following special functions are also involved in the findings of this paper.

Definition 4.
e beta function, also referred to as first type of Euler integral, is defined by where Re(α), Re(s) > 0.
Close association of the beta function to the gamma function is an important factor of the beta function e beta function is symmetric, i.e., β(α, s) � β(s, α). A generalization of the beta function, called the incomplete beta function, is defined by where Re(α), Re(s) > 0 with 0 < b < 1. e incomplete beta function β(b; α, s) weakens to the ordinary β(α, s) (beta function) by setting b � 1.
In [8], the Hadamard inequality is studied for Riemann-Liouville fractional integrals which is stated in the following theorem.
with α > 0. e inequality stated in the aforementioned theorem motivates the researchers to work in this direction by establishing other kinds of inequalities for Riemann-Liouville fractional integrals. In the past decade, several classical inequalities have been extended via different kinds of fractional integral operators.
is paper is organized as follows. In Section 2, two versions of the Hadamard inequality for strongly (s, m)-convex functions via Riemann-Liouville fractional integrals are given. eir connection with the well-known results is established in the form of corollaries and remarks. In Section 3, the error estimations of Hadamard inequalities for Riemann-Liouville fractional integrals are obtained by using differentiable strongly (s, m)-convex functions.

Main Results
, then the following fractional integral inequality holds: with α > 0.
Proof. Since f is strongly (s, m)-convex function, for By multiplying inequality (11) with t α− 1 on both sides and then integrating over the interval [0, 1], we get By change of variables, we will get Further, the above inequality takes the following form: From the definition of strongly (s, m)-convex function with modulus C, for t ∈ [0, 1], we have the following inequality: By multiplying inequality (15) with t α− 1 on both sides and then integrating over the interval [0, 1], we get

Journal of Mathematics
By change of variables, we will get Further, the above inequality takes the following form: From inequalities (14) and (18), one can get inequality (9).
Corollary 2. For α � 1 and m � 1, the following inequality holds for strongly s-convex function: In the next theorem, we give another version of the Hadamard inequality.

Proof. Let t ∈ [0, 1]. Using strong (s, m)-convexity of function f for u � x(t/2) + m((2 − t)/2)y and v � ((2 − t)/2)(x/m) + y(t/2) in inequality
By multiplying (22) with t α− 1 on both sides and making integration over [0, 1], we get By using change of variables and computing the last integral, from (23), we get Further, it takes the following form: e first inequality of (21) can be seen in (25). Now we prove the second inequality of (21). Since f is strongly (s, m)-convex function and t ∈ [0, 1], we have the following inequality: Journal of Mathematics 5 (26) By multiplying inequality (26) with t α− 1 on both sides and making integration over [0, 1], we get By using change of variables and computing the last integral, from (27), we get Further, it takes the following form: Corollary 4. For m � 1 in (21), we get the result for Riemann-Liouville fractional integrals of strongly s-convex function: Corollary 5. For m � 1 and C � 0 in (21), we get the result for Riemann-Liouville fractional integrals of s-convex function: Corollary 6. For m � 1 and α � 1 in (1), we have the Hadamard inequality for strongly s-convex function:

Error Estimations of Riemann-Liouville Fractional Integral Inequalities
e following two lemmas are very useful to obtain the results of this section.
By using Lemma 1 and (37), we have After simplifying the last inequality of (38), we get (36).