Generalized Integral inequalities for Hermite-Hadamard type via s-convexity on fractal sets

In this article, the new HermiteHadamard type inequalities are studied via generalized s-convexity on fractal sets. These inequalities derived on fractal sets are shown to be the generalized s-convexity on fractal sets. We proved that the absolute values of the first and second derivatives for the new inequalities are the generalization of s-convexity on fractal sets.


Introduction
The convexity is considered among the important property in mathematical analysis.The applications of convex functions can be found in many fields of studies including economy, engineering and optimization (see for example [1,4]).A well-known result which was identified as Hermite-Hadamard inequalities is the reformulation through the convexity.These inequalities, widely reported in the literature, can be defined as follows: These two inequalities, which are the refinement to the convexity, can be held in reverse order as concave.Following this, many refinements of convex functions using Hermite-Hadamard inequalities have been continuously studied [3,5,7].Given the variation of Hermite-Hadamard inequalities, Dragomir and Fitzpatrick [6] established a new generalizations 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 Even though the Hermite-Hadamard inequalities are established for classical integrals, the inequalities can also hold for fractional calculus.Other important generalizations include the work of Sarkaya et al. [8], who proved the Hermite-Hadamard inequalities through fractional integrals as follows: we have: where α ≥ 0.
The s-convexity mentioned in Hudzik and Maligranda [12] was also given as the generalization on fractal sets.
The Riemann-Liouvile fractional integral is introduced here due to its importance.
The Riemann-Liouvile integrals J α u+ ψ and J α v− ψ of order α ∈ R + are defined by and The following lemma for differentiable mapping is given by Sarikaya et al. [8].
then we have: Wang et al. [11] extended Lemma 1 to include two casses, one of wich involves the second derivative of Riemann-Liouvile fractional integrals.
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 generalized form of Theorem 2. The new HermiteHadamard type inequalities in the class of functions having 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 by 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: where Using inequalities (7) and (11), we proved Theorem 4. .
(iii) Applying the second generalized Hermite-Hadamard inequalities, we obtain Note that if γ = 0, then the inequality holds as it is equivalent to and we know that for s ∈ (0, 1).Since and Forall γ ∈ [0, 1] and x ∈ [u, v], then we obtain and the inequality (13) is proved.
(iv) We have and and the proof of Theorem 5 is complete.
Corollary 2. Choosing s = 1 in Theorem 5, we have we get For some fixed q ≥ 1, if |ψ | q is a generalized s-convex on (u, v), we obtain Proof.Applying Lemma 1, we obtain First, suppose q = 1.Since the function |ψ (u)| is a generalized s-convex on (u, v), we obtain Therefore, Suppose that q > 1, from the power mean inequality In view of inequalities ( 14), ( 16) and ( 17) complete the proof of Theorem 6.
Corollary 3.Under the similar conditions of Theorem 6, we get (iii) If q > 1 and s = 1 From this fact and applying the Hlder's inequality, we have Thus, the inequalities ( 14) and (18) complete the proof of Theorem 7.
Remark 3. From Theorem 8, 6 and 7 we obtain the following inequality for q > 1 where For some fixed q ≥ 1, if |ψ | q is a generalized s-convex on (u, v), then we get Proof.Applying Lemma 2, we have Firstly, suppose that q = 1.Since the mapping |ψ | is generalized s-convexity on fractal sets, we obtain Therefore, where, Secondly, for q > 1.From Lemma 2 and the power mean inequality, we have Hence, from inequalities (21) and ( 22), we obtain , where 0 ≤ u < v, s ∈ (0, 1] and for some fixed q > 1, then we get , where 1 p + 1 q = 1. Proof.From (20), (21) and the Hlders inequality, we have for any γ ∈ [0, 1], which follows from where V > N ≥ 0 and q ≥ 1.
The proof of Theorem 10 is completed.
The following another Hermit-Hadamard type inequalities of the second derivatives.
Using the results obtained in Section 2, and the above applications of means, we get the following proposition.
Proof.This follows from Corollary 3 (ii) wich applied for ψ(x) = x r , we get the required result.
Proof.This follows from Corollary 3 (iv) wich applied for ψ(x) = x n , we get the required result.
Proposition 3. Suppose that u, v ∈ R such that 0 < u < v, then Proof.This follows from Corollary 3 (ii) wich applied for ψ(x) = 1 x , we obtain the required result.