Integral inequalities for some convex functions via generalized fractional integrals

In this paper, we obtain the Hermite–Hadamard type inequalities for s-convex functions and m-convex functions via a generalized fractional integral, known as Katugampola fractional integral, which is the generalization of Riemann–Liouville fractional integral and Hadamard fractional integral. We show that through the Katugampola fractional integral we can find a Hermite–Hadamard inequality via the Riemann–Liouville fractional integral.


Introduction
A function f : I → R, where I is an interval of real numbers, is called convex if the following inequality holds: for all a, b ∈ I and t ∈ [0, 1]. Function f is called concave if -f is convex. The Hermite-Hadamard inequality [4] for convex functions f : I → R on an interval of real line is defined as where a, b ∈ I with a < b.
for all a, c ∈ [0, b] and t ∈ [0, 1] and for all m The left-and right-hand side Riemann-Liouville fractional integrals of order α of function f are given by respectively, where (α) is the gamma function defined by (α) = ∞ 0 e -t t α-1 dt.

Definition 1.4 ([16])
Let α > 0 with n -1 < α ≤ n, n ∈ N, and 1 < x < b. The left-and right-hand side Hadamard fractional integrals of order α of function f are given by and where ess sup stands for essential supremum.
In this paper, we give the Hermite-Hadamard type inequalities for s-convex functions and for m-convex functions via generalized fractional integral. Throughout the paper, X p c (a, b) (c ∈ R, 1 ≤ p ≤ ∞) is the space as defined in Definition 1.5 and L 1 [a, b] stands for the space of Lebesgue integrable over the closed interval [a, b] where a, b are some real numbers with a < b.

Hermite-Hadamard type inequalities for s-convex function
In this section we give Hermite-Hadamard type inequalities for s-convex function.
, then the following inequalities hold: where the fractional integrals are considered for the function f (x ρ ) and evaluated at a and b, respectively.
Then we have Multiplying both sides of (6) by t αρ-1 , α > 0 and then integrating the resulting inequality with respect to t over [0, 1], we obtain This establishes the first inequality. For the proof of the second inequality in (5), we first observe that for an s-convex function f , we have By adding these inequalities, we get Multiplying both sides of (8) by t αρ-1 , α > 0 and then integrating the resulting inequality with respect to t over [0, 1], we obtain and by choosing the change of variable t ρ = z, we have Thus (9) becomes Thus (7) and (10) give (5).
, then the following inequality holds: Proof From (7) one can have Integrating by parts, we get By using the triangle inequality and s-convexity of |f | and the change of variable t ρ = z, we obtain Corollary 2.4 Under the same assumptions of Theorem 2.3.

Lemma 2.5 Let α > 0 and
Then the following equality holds if the fractional integrals exist: Proof By using the similar arguments as in the proof of Lemma 2 in [18]. First consider Similarly, we can show that Thus from (19) and (20) we get (18).
Throughout all other results we denote for some fixed q ≥ 1, then the following inequality holds: Proof Using Lemma 2.5 and the power mean inequality and s-convexity of |f | q , we obtain Hence the proof is completed.

Corollary 2.8
Under the similar conditions of Theorem 2.7.
for some fixed q ≥ 1, then the following inequality holds: Proof Using Lemma 2.5, the property of modulus, the power mean inequality, and the fact that |f | q is an s-convex function, we have By using the change of variable t ρ = z, we get Thus substituting the values of A and B in (24) and applying the fact that β(a, b) = β(b, a), we get the desired result.

Corollary 2.10
Under the similar conditions of Theorem 2.7.
Then we have Proof Since f is an m-convex function, we have and also Take Multiplying both sides of (33) by t αρ-1 , α > 0 and then integrating the resulting inequality with respect to t over [0, 1], we obtain Then, by the change of variable u ρ = t ρ a ρ + (1t ρ )b ρ , we get the desired inequality (32).

Applications to special means
In this section, we consider some applications to our results. Here we consider the following means: (1) The arithmetic mean: (2) The logarithmic mean: L(a, b) = ln |b| -ln |a| ba ; a, b ∈ R, |a| = |b|, a, b = 0.
Proof By taking f (x) = x n in Corollary 2.4(3), we get the required result.
Proof By taking f (x) = x n in Corollary 2.8(3), we get the required result.
Proof By taking f (x) = x n in Corollary 2.10(3), we get the required result.