On Hadamard's inequalities for the convex mappings defined in topological groups and a connected result

Abstract

In this paper, we study the Hadamard's inequality for midconvex and quasi-midconvex functions in topological groups. A mapping naturally connected with this inequality and related result is also pointed out.

On Hermite-Hadamard type inequalities for the product of two convex mappings defined on topological groups

“…Now we prove Hermite-Hadamard type inequalities for products of midconvex and quasi-convex functions defined in a topological groups. Before we proceed to prove our results we give some definitions from [1,11,12].…”

“…Definition 3.5. [12] The function f : Ω → R is said to be a P-function in Ω, if f (a) f (az) + f (az −1 ) for all a ∈ Ω and z ∈ G such that az, az −1 ∈ Ω.…”

“…In [12], Morassaei established the Hermite-Hadamard type inequality for midconvex and quasi-midconvex functions in topological groups and discussed some of the properties of the mapping naturally connected with the Hermite-Hadamard inequality for globally midconvex functions defined in a topological group. Some of the main results of [12] are stated in the following theorems. Theorem 1.1.…”

“…Theorem 1.2. [12] Suppose that G is a locally compact group and Ω ⊂ G an open symmetric set relative to a ∈ G with 0 < µ(Ω) < ∞ and e ∈ Ω. Let f be measurable and quasi-midconvex real-valued function on Ω such that f ∈ L 2 (Ω).…”