Characterization of approximately monotone and approximately Hölder functions

Abstract

A real valued function f defined on a real open interval I is called Φ-monotone if, for all x,y ∈ I with x y it satisfies f (x) f (y) + Φ(y − x), where Φ : [0, (I)[→ R + is a given nonnegative error function, where (I) denotes the length of the interval I. If f and − f are simultaneously Φ-monotone, then f is said to be a Φ-Hölder function. In the main results of the paper, using the notions of upper and lower interpolations, we establish a characterization for both classes of functions. This allows one to construct Φmonotone and Φ-Hölder functions from elementary ones, which could be termed the building blocks for those classes. In the second part, we deduce Ostrowski-and Hermite-Hadamardtype inequalities from the Φ-monotonicity and Φ-Hölder properties, and then we verify the sharpness of these implications. We also establish implications in the reversed direction.

On approximately convex and affine functions

“…In what follows, we are going to define four properties related to an error function Φ ∈ E(I). First we recall the notions of Φ-monotone and Φ-Hölder functions that have been introduced in our former papers [10,11].…”

Hermite–Hadamard-Type Inequalities and Two-Point Quadrature Formula

“…They showed that a function satisfying δ-convexity can be decomposed as the algebraic sum of an ordinary convex and a bounded function whose supremum norm is not greater than δ 2 . Since then many different versions of approximate convexity were introduced and investigated (see [2], [3], [4] and their references).…”

“…This version of approximate monotonicity was first introduced in [2]. A more in depth study can be found in [3]. Due to association of the non-negative error term, the class of Φ-monotone functions is bigger than the class of ordinary nondecreasing functions.…”