On approximately Convex and Affine Sequences

Abstract

In this paper, our primary objective is to study a possible decomposition of an approximately convex sequence. For a given $\varepsilon>0$; a sequence $\big<u_n\big>_{n=0}^{\infty}$ is said to be $\varepsilon$-convex, if for any $i,j\in\mathbb{N}$ with $i<j$ there exists an $n\in]i,j]\cap \mathbb{N}$ such that the following discrete functional inequality holds \begin{equation*} { u_i-u_{i-1}-\dfrac{\varepsilon}{n-i}\leq u_j-u_{j-1}. } \end{equation*} We show that such a sequence can be represented as the algebraic summation of a convex and a controlled sequence which is bounded in between $\left[-\dfrac{\varepsilon}{2}, \dfrac{\varepsilon}{2}\right].$ On the other hand, if for any $i,j\in\mathbb{N}$ with $i<j$, if a sequence $\big<u_n\big>_{n=0}^{\infty}$ satisfies the following form of inequality \begin{equation*} { \left|\big(u_i-u_{i-1}\big)-\big(u_j-u_{j-1}\big)\right|\leq\dfrac{\varepsilon}{n-i}\quad \quad\mbox{for some} \quad n\in]i,j]\cap\mathbb{N}; } \end{equation*} then we term it as $\varepsilon$-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 $\varepsilon.$