On Some Hadamard-Type Inequalities for Convex Functions On A Rectangular Box

Abstract

In this paper, a few Hadamard-type inequalities for convex functions of 3−variables on a rectanguler box are defined. Mapping related to convex functions on a rectanguler box are also described.

Hermite-Hadamard Inequality for Geometrically Quasiconvex Functions on a Rectangular Box

“…is known in the literature as Hermite-Hadamard inequality for convex functions (see for example [5]). In resent years there have been many extensions, generalizations, refinements and similar results of Hetmite-Hadamard inequality, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and references therein.…”

“…Furthermore, there exists co-ordinated convex function which is not convex (see [11]). Since then several important generalizations introduced on this category, see [14,15,[18][19][20] and references therein.…”

“…On the other hand, M.E.Özdemir [15] defined convex functions on a rectangular box Ω := [a, b] × [c, d] × [e, f ] in R 3 as follows: A function g : Ω → R is said to be convex on the co-ordinates on Ω if for every (x, y, z) ∈ Ω, the partial mapping, Theorem 1.1. Suppose that g : Ω → R is convex function.…”

“…g z : [a, b] × [c, d] → R, g z (u, v) = g(u, v, z), z ∈ [e, f ], g y : [a, b] × [e, f ] → R, g y (u, w) = g(u, y, w), y ∈ [c, d], g x : [c, d] × [e, f ] → R, g x (v, w) = g(x, v, w), x ∈ [a, b],are convex. Recall the following inequality of Hermite-Hadamard type for coordinated convex function on a rectangular box in R 3 from[15].…”