Integral inequalities for s-convexity via generalized fractional integrals on fractal sets

In this study, we establish a new integral inequalities of Hermite-Hadamard type for $s$-convexity via Katugampola fractional integral. This generalizes the Hadamard fractional integrals and Riemann-Liouville into a single form. We show that the new integral inequalities of Hermite-Hadamard type can be obtained via the Riemann-Liouville fractional integral. Finally, we give some applications to special means.


Introduction
Fractional calculus, whose applications can be found in many fields of studies including economics, life and physical sciences, as well as engineering, can be considered as one of the modern branches of mathematics [1,2,3] and [4].Numerus problems of interests from these fields can be analyzed through the fractional integrals, which can also be regarded as an interesting sub-discipline of fractional calculus.Some of the applications of integral calculus can be seen in the following papers [5,6,7,8], through which problems in physics, chemistry and population dynamics were studied.The fractional integrals were extended to include the Hermite-Hadamard type inequalities, which are classically given as follows.
Consider a convex function, ψ : E ⊆ R → R , u, v ∈ E with u ≤ v, if and only if, Following this, many important generalizations of Hermite-Hadamard inequality were studied [9,10,11,12,13,14], some of which were formulated via generalized s-convexity, which is defined as follows.
This class of function is denoted by GK 2 s (see [15]).
In [16] Mehran and Anwar studied the Hermite-Hadamard-type inequalities for s-convexity involving generalized fractional integrals.The following definitions of the generalized fractional integrals were given in [17].
For order α > 0, the tow sides of Katugampola fractional integrals for ψ ∈ X p c (u, v) are defined by When improving the results in [16], Mehran and Anwar used Definition 2 together with the following lemma.
, where 0 ≤ u < v for α > 0 and ρ > 0. If the fractional integrals exist, we get This paper is aimed at establishing some new integral inequalities for generalized s-convexity via Katugampola fractional integrals on fractal sets linked with (1).Some inequalities are presented here for the class of mappinges which their derivatives in absolute value at certin powers are generalized s-convexity.Also, we obtained some new inequalities linked with convexity and generalized s-convexity via classical integrals as well as Riemann-Liouville fractional integrals in form of a Corollary.As an application, the inequalities for special means are derived.

Main results
Hermite-Hadamard inequalites for s-convexity via a generalized fractional integral can be wriiten with the aid of the following theorem.
then we obtain (2) Combining the above inequalities, we have Multiplying both sides of inequality (3) by t αρ−1 , for α > 0 and integrating it over [0, 1] with respect to t, we obtain Since , and applying change of variable t ρ = a, we get Thus, inequality (4) becomes When proving the first part of inequality (2), we observe that ψ is a generalized s-convex function on [u ρ , v ρ ], through wich the following inequality is obtained for x, y ∈ [a, b], α ≥ 0. Consider , where t ∈ [0, 1].Applying inequality (5), we have Multiplying both sides of the inequality (6) by t αρ−1 , for α > 0 and then integrating over [0, 1] with respect to t give the following Then it follows that where β(u, v) is the Beta function.
Remark 1.When substituting ρ = 1 and α = 1 in Theorem 1, we reported the results of this theorem to that of Theorem 2.1 (see Dragomir and Fitzpatrick [9]).
In the next theorem, the new upper bound for the right-hand side of (1) for generalized sconvexity is proposed.For this recall that the generalized beta function is defined as Proof.In view of Lemma 1, we have First, suppose that q = 1.Since |ψ ′ | is a grneralized s-convex on [a ρ , b ρ ], we have Therefore, Calculating S 1 and S 2 , we get and Thus, if we use inequalities ( 10) and ( 11) in ( 9), we obtain The inequalities ( 8) and ( 12) complete the proof for this case.Consider the second case, q > 1.Using inequality (8) and the power mean inequality, we obtain The inequalities ( 8) and ( 13) complete the proof.
Corollary 1.Using the similar assumptions given in Theorem 2.
The other type is given by the next theorem.
Proof.This follows from Corollary 2 applied for ψ(x) = x n , we get the required result.
Proposition 3. Suppose that u, v ∈ R, where 0 < u < v. Then for q ≥ 1, we get Proof.This follows from Corollary 1(iii) applied for ψ(x) = 1 x , we get the required result.Proposition 4. Suppose that u, v ∈ R, where 0 < u < v. Then for q ≥ 1, we get Proof.This follows from Corollary 2 applied for ψ(x) = 1 x , we get the required result.