Generalized Integral Inequalities for Hermite–Hadamard-Type Inequalities via s-Convexity on Fractal Sets

In this article, we establish new Hermite–Hadamard-type inequalities via Riemann–Liouville integrals of a function ψ taking its value in a fractal subset of R and possessing an appropriate generalized s-convexity property. It is shown that these fractal inequalities give rise to a generalized s-convexity property of ψ. We also prove certain inequalities involving Riemann–Liouville integrals of a function ψ provided that the absolute value of the first or second order derivative of ψ possesses an appropriate fractal s-convexity property.


Introduction
Convexity is considered to be an important property in mathematical analysis. The applications of convex functions can be found in many fields of studies including economics, engineering and optimization (see for example [1,2]). A well-known result which was identified as Hermite-Hadamard inequalities is the reformulation through convexity. These inequalities, widely reported in the literature, can be defined as follows: These two inequalities, which are refinement of convexity, can be held in reverse order as concave. Following this, many refinements of convex functions using Hermite-Hadamard inequalities have been continuously studied [3][4][5][6]. Given the variation of Hermite-Hadamard inequalities, Dragomir and Fitzpatrick [7] established a new generalization of s-convex functions in the second sense.
Theorem 2. Suppose that ψ : R + → R + is a s-convex function in the second sense, where 0 < s ≤ 1, u, v ∈ R + and u < v. If ψ ∈ L 1 ([u, v]), then Though the Hermite-Hadamard inequalities were established for classical integrals [8], the inequalities can also hold for fractional calculus, such as Riemann-Liouville [9][10][11], Katugampola [12] and local fractional integrals [13]. Some of these were studied through Mittag-Leffler function [14,15]. Other important generalizations include the work of Sarikaya et al. [16], who proved the Hermite-Hadamard inequalities through fractional integrals as follows: The s-convexity mentioned in Hudzik and Maligranda [2] was also given as the generalization on fractal sets.
The Riemann-Liouville fractional integral is introduced here due to its importance. respectively.
The following lemma for differentiable function is given by Sarikaya et al. [16].
Wang et al. [9] extended Lemma 1 to include two cases, one of which involves the second derivative of Riemann-Liouville fractional integrals.

holds.
Even though studies were conducted on generalized Hermite-Hadamard inequality via Riemann-Liouville fractional integrals for s-convexity [16,[19][20][21], inequalities of this type for generalized s-convexity are lacking. Therefore, this paper is aimed at establishing some new integral inequalities via generalized s-convexity on fractal sets. We show that the newly established inequalities are generalizations of Theorem 2. The new Hermite-Hadamard-type inequalities in the class of functions with derivatives in absolute values are shown to be s-convex function on fractal sets. This was achieved using Riemann-Liouville fractional integrals inequalities.

Main Results
Our first main result is obtained in the following theorem.
To prove the second inequality in (4), since ψ ∈ GK 2 s , we get and Combining the inequalities (8) and (9), we obtain A similar technique used in (6) is applied to inequality (10) to get the following: Using inequalities (7) and (11), we prove Theorem 4.

Remark 1.
In the second inequality of Theorem 4, the expression 1 given by ψ(z) = z sα is generalized s-convex in the second sense, and it satisfies the following equalities:
This result is the same as Theorem 2.1 in Dragomir and Fitzpatrick [7].

Remark 2. The equality
Theorem 5. Suppose that M : [0, 1] → R α is the mapping given by We have the following inequality: (iii) We have the following inequality: where Proof.
(ii) Assume that γ ∈ (0, 1]. Then by the change of variables q = γv + (1 − γ) u+v 2 and p = γu Applying the first generalized Hermite-Hadamard inequality, we obtain and inequality (12) is obtained. If γ = 0, the inequality also holds. (iii) Applying the second generalized Hermite-Hadamard inequality, we obtain Please note that if γ = 0, then the inequality holds as it is equivalent to which is known to hold for s ∈ (0, 1].
Since for all γ ∈ [0, 1] and x ∈ [u, v] the inequalities and are true, we obtain and the inequality (13) is proved. (iv) We have Since and the proof of Theorem 5 is complete.

Corollary 2.
Choosing s = 1 in Theorem 5, we have (i) Proof. Applying Lemma 1, we obtain First, suppose q = 1. Since the function |ψ | is generalized s-convex on (u, v), we obtain Therefore, . (16) Next suppose that q > 1. From the power mean inequality and the generalized s-convexity of the function |ψ | q we obtain In view of inequalities (14), (16) and (17) the proof of Theorem 6 is complete now.
(ii) If q = α = s = 1, then (iii) If q > 1 and s = 1 (iv) If q > 1 and α = s = 1 then If |ψ | q is generalized s-convex on (u, v) for q > 1, we get Proof. Since |ψ | is generalized s-convex on (u, v), we obtain From this fact and applying the Hölder's inequality, we have Thus, the inequalities (14) and (18) complete the proof of Theorem 7.
Secondly, for q > 1. From Lemma 2 and the power mean inequality, we have Hence, from inequalities (21) and (22), we obtain This completes the proof of Theorem 9.
The proof of Theorem 10 is complete now.
The following result exhibits another Hermite-Hadamard type inequality in terms of the second derivative of a function.
Theorem 11. Under the same assumptions of Theorem 10, we have Proof. By applying Lemma 2 and the Hölder's inequality, we obtain This completes the proof of Theorem 11. Remark 4. From Theorems 9, 10 and 11, we have Proposition 4. Suppose that u, v ∈ R such that 0 < u < v, then Proof. This result follows from Corollary 3 (iv) applied to the function ψ(x) = x −1 .
Funding: This research received no external funding.