Some generalized ostrowski type fractional integral inequalities for MT − convex functions with applications on special means

Some generalized Ostrowski-type integral inequalities for r − times di ﬀ erentiable functions whose absolute values are MT − convex have been discussed. Moreover, some applications on special bivariate means are obtained.


Introduction
In 1938, A. M. Ostrowski proved an interesting integral inequality, estimating the absolute value of the derivative of a differentiable function by its integral mean as follows: Theorem 1. [3] Let f : K → R, K is an interval in R, be a differentiable function in K o , the interior of K and a, b ∈ K o , a < b. If |f 0 (t)| ≤ M for all t ∈ [a, b], then Inequality (1.1) has great importance in numerical analysis and in the theory of some special means, it attracts the attention and interest of mathematicians and researchers. It was generalized, extended and different variants were given. These studies include, among others, the works in [1,2,12,15]. The development of the theory of fractional calculus is prior to the turn of 20 th century. It has a lot of applications in different fields of science and engineering including viscoelastic materials, fluid flow, rheology, diffusive transport, electrical networks, electromagnetic theory, probability. Fractional calculus is the generalization of classical calculus. Like ordinary calculus, fractional derivatives and integrals are not defined in a unique way. Different authors have their contributions [6,7,11,14]. For n = 1, the sum P n−1 k=1 is vacuously considered to be zero. R is the set of real numbers. In the whole presentation all the integrals are finite while, I := [a ρ , (a + η(b, a)) ρ ] is an interval in R, U depicts a convex set in R and η : M × M → R a continuous function where M ⊆ R. This work is organized in the following way. After this Introduction, in Section 2 some basic concepts are discussed. In Section 3 some results relating to the topic are established and Section 4 deals with some applications on special means.

Preliminaries
Definition 1. [10] The left-and right-sided Riemann-Liouville fractional integrals of order α > 0 of f, denoted by J α a+ f and J α b− f respectively, are defined as: where, Γ is the Euler gamma function; α → 1 gives the classical integral and it may be noted that: Definition 2. [17] The left-and right-sided Hadamard fractional integrals of order α > 0 of f, denoted by H α a+ f and H α b− f respectively, are defined as: Recently Katugampola defined the following integrals generalizing both Riemann-Liouville and Hadamard fractional integrals.
[10] Let f : U → [0, ∞) be a function, then f is said to be convex (or that f ∈ Conv(U )), provided that: The ρ−gamma and ρ−Beta functions for any two positive real numbers x, y are, denoted by ρ Γ(x) and ρ B(x, y) respectively, defined as [4]: interior of I, and f (r) ∈ X p c (a, a+η(b, a)) for r ∈ N 0 ; let β := α−j +1+ 1−n ρ for r ≥ j; ρ, α, β > 0 and n ∈ N. Moreover, if¯f (r)¯∈ MT (I) is such that Proof. Applications of Lemma 1, boundedness, MT −convexity of¯f (r)ā nd relation (2.7) yield: This completes the proof. 2 Corollary 1. Let the conditions of Theorem 1 be satisfied for Theorem 2. Let the conditions of Theorem 1 be satisfied; Moreover, if p, q > 1 for which p = q q−1 and¯f (r)¯q ∈ MT (I), then Proof.
, yields the desired result.  + η(b, a) (3.5) Corollary 3. Let the conditions of Theorem 2 be satisfied for x = a and x = a + η(b, a), then Theorem 3. Let the conditions of Theorem 1 be satisfied; Moreover, if q ≥ 1 for which¯f (r)¯q ∈ MT (I), then

Applications
Let a, b > 0 with a 6 = b, then the arithmetic mean and p−logarithmic mean on a, b is defined as: Proposition 1. Let f : (0, 1] → R, be defined by f (x) = − ln x s where s > 0, obviously f is MT −convex and hence from inequality (3.1), the following inequality holds: + η(b, a)), 2a + η(b, a))] j+β−k−1 where, B is the Incomplete Beta function.