Some Hermite-Hadamard type inequalities for MT-convex functions via classical and Riemann-Liouville fractional integrals

Abstract

Some inequalities of Hermite-Hadamard type for MT-convex functions via classical integrals and Riemann-Liouville fractional integrals are introduced, respectively.

Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function

“…In particular, if ( ) = ρ ζ ζ , then we have the following corollary, which has been proved by Park [29].…”

“…By virtue of the concept of MT-convexity, Park in [29] proved the following Hermite-Hadamard-type inequalities. , then for all ∈ ( ) h 0, 1 and > μ 0 we have…”

Hermite‐Hadamard inequalities for Riemann‐Liouville fractional integrals of a convex function with respect to a monotone function

“…With the similar assumptions of Lemma 1 if we take (x) = x, then inequalities (6) reduce to inequalities (3). Moreover, if we take (x) = x and = 1, then inequalities (6) reduce to inequalities (2).…”

“…If we take = 1 in (3), we obtain (2), so it is clear that inequality (3) is a generalization of Hermite-Hadamard inequality (2). Many inequalities have been established in view of inequality (3) for convex functions, 1 h-convex functions, 2 MT-convex functions, 3,4 m-convex functions, 12 (s, m)-convex functions, 12 ( , m)-convex functions, 13,24 F-convex functions, 14 and invex and preinvex functions, 15,16 and other kinds of convex functions can be found in 17,18 and related references therein.…”

New Hermite-Hadamard inequalities for twice differentiable ϕ -MT-preinvex functions

“…For some interesting and significant integral inequalities concerning with the MT-convex functions, one can see in the recent papers [14,21,[27][28][29]. It can be easily observed that convexity means just Jensen-convex when t = In [18],Özdemir et al established inequalities for twice differentiable m-convex functions which are connected with Hermite-Hadamard inequality, and they used the following lemma to prove their results.…”