P-functions, Quasi-convex Functions, and Hadamard-type Inequalities

Abstract

We establish some results concerning P-functions from the standpoint of abstract convexity. In particular, we show that the set of all P-functions on a segment is the least set closed under pointwise sum, supremum, and convergence and containing the class of all nonnegative quasi-convex functions on that segment. Further, generalizations are derived of a recent inequality of Hadamard type for P-functions.

Inequalities of Hermite-Hadamard Type for Composite <em>h</em>-Convex Functions

“…Obviously Q (I) contains P (I) and for applications it is important to note that also P (I) contain all nonnegative monotone, convex and quasi convex functions, i. e. nonnegative functions satisfying (1.3) f (tx + (1 t) y) max ff (x) ; f (y)g for all x; y 2 I and t 2 [0; 1] : For some results on P -functions see [45] and [62] while for quasi convex functions, the reader can consult [44].…”

On Quasi Convex Functions and Hadamard's Inequality

“…(see [8,11,12,14]) We say that f : I → R is a P −function, or that f belongs to the class P (I), if f is nonnegative and for all x, y ∈ I and λ ∈ [0, 1], we have f (λx + (1 − λ)y) ≤ f (x) + f (y).…”

Generalization and reverses of the left Fejer inequality for convex functions

“…For related results, see for instance the research papers [1,[4][5][6][7][8][9][10][11], the monograph online [2] and the references therein.…”