Hermite–Hadamard-type inequalities involving ψ-Riemann–Liouville fractional integrals via s-convex functions

In this paper, we establish some new Hermite–Hadamard-type inequalities involving ψ-Riemann–Liouville fractional integrals via s-convex functions in the second sense. Meanwhile, we present many useful estimates on these types of new Hermite–Hadamard-type inequalities. Finally, we give some applications to special means of real numbers.


Introduction
The classical Hermite-Hadamard inequality is as follows: for convex functions g : [a, b] ⊂ R → R (see [1]). In the past decade, fractional calculus has been regarded as one of the best tools to describe long-memory processes. Many researchers are interested in such a model. The most important of these models are described by differential equations with fractional derivatives. Their evolution is much more complex than the classical integer-order case, and the corresponding theory is also more difficult in the integer-order case. The theory of fractional integral inequalities plays an important role in mathematics.
With a wide application of fractional integration and Hermite-Hadamard inequality, many researchers extended their research to the Hermite-Hadamard inequality, including fractional integration rather than ordinary integration; see [19][20][21][22][23][24][25][26][27]. Sarikaya et al. [19] derived an interesting Hermite-Hadamard-type inequality, which contains the fractional integral instead of the ordinary one. The study attracted many researchers to consider the problem. So far, some new integral inequalities have been obtained by using fractional calculus. Sousa et al. [28] introduced fractional integral operators with ψ-Riemann-Liouville kernel and proved similar inequalities.
In addition to the classical convex functions, Hudzik and Maligranda [29] introduced the definition of s-convex functions in the second sense.
for all x, y ∈ I and λ ∈ [0, 1] and for some fixed s ∈ (0, 1]. be a finite or infinite interval of the real line R, and let α > 0. Also, let ψ(x) be an increasing positive function on (a, b] with continuous derivative ψ (x) on (a, b). Then the left-and right-sided ψ-Riemann-Liouville fractional integrals of a function f with respect to the function ψ on [a, b] are defined by respectively, where Γ is the gamma function.
Then we have the following equality for fractional integrals: Proof From [31] we have The proof is completed.
then we have the following equality for fractional integrals: Proof Note that By Lemma 1.3 we have The proof is completed.
In the case where Ω is strictly convex on I, we have equality in (4) if and only if f is constant almost everywhere on Ω.
The main purpose of this paper is to introduce some new Hermite-Hadamard-type inequalities involving ψ-Riemann-Liouville fractional integrals via s-convex functions in the second sense. For these functions, we establish some results related to the left end of new inequalities similar to inequality (1). We give some applications to special mean of a positive real number.

Main results
We now in a position to establish some inequalities of Hermite-Hadamard type involving ψ-Riemann-Liouville fractional integrals (with α ∈ (0, 1)) via s-convex functions.

c], and let ψ be an increasing positive function on [b, c] having a continuous derivative ψ on (b, c). If h is an s-convex function on [b, c], then we have the following inequality for fractional integrals:
Multiplying both sides of (7) by t α-1 and then integrating the resulting inequality with respect to t over [0, 1], we obtain Next, so the left-hand side inequality in (6) is proved.
To prove the right-hand side inequality in (6), since h is an s-convex function, for t ∈ [0, 1], we have and then Multiplying both sides of (8) by t α-1 and then integrating, we obtain

So then
The proof is completed.
Proof Using Lemma 1.4 and the s-convexity of h, we have The proof is completed.
Proof Using Lemma 1.4 and the Hölder inequality via the s-convexity of |h | q (q > 1), we have The proof is completed.
Proof We consider inequality (10), and we let a . . , n, we obtain the required result. This completes the proof.  (b, c). If |h | q (q > 1) is an s-convex function on [b, c] for some fixed s ∈ (0, 1], then we have the following inequality for fractional integrals: Proof Using Lemma 1.4 and the power mean inequality via the s-convexity of |h | q (q > 1), we have The proof is completed.
Proof We can obtain the result using the technique in the proof of Corollary 2.4 by considering inequality (13).