On approximately convex and affine functions

Abstract

In this paper, our primary objective is to study a possible decomposition of an approximately convex sequence.For a given ε > 0; a sequence un ∞ n=0 is said to be ε-convex, if for any i, j ∈ N with i < j there exists an n ∈]i, j] ∩ N such that the following discrete functional inequality holdsWe show that such a sequence can be represented as the algebraic summation of a convex and a controlled sequence which is bounded in between − ε 2 , ε 2 .On the other hand, if for any i, j ∈ N with i < j, if a sequence unthen we term it as ε-affine sequence. Such a sequence can be decomposed as the algebraic summation of an affine and a bounded sequence whose supremum norm doesn't exceed ε.Various definitions, backgrounds, motivations, findings, and other important things are discussed in the introduction section.

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).…”

“…With the help of error function Φ, we can introduce terminologies such as approximately monotone, Hölder, convex and affine functions. These function classes are studied in depth in the papers [2]- [4], [6], where along with structural properties, some characterizations, decompositions, several sandwich type theorems and applications are discussed. For readability purposes, we recall these notions.…”