Decomposition of Approximately Monotone and Convex Sequences

TL;DR

This paper explores how approximately monotone and convex sequences can be decomposed into simpler components, introduces a new operator for convex functions, and studies their properties and relationships.

Contribution

It provides novel decomposition methods for approximately monotone and convex sequences and introduces a twisting operator with applications to convex functions.

Findings

Any sequence can be expressed as the difference of two nondecreasing sequences.

Approximately monotone sequences can be closely approximated by non-decreasing sequences.

Approximately convex sequences can be written as the difference of a convex sequence and a Lipschitz sequence.

Abstract

In this paper, we primarily deal with approximately monotone and convex sequences. We start by showing that any sequence can be expressed as the difference between two nondecreasing sequences. One of these two monotone sequences act as the majorant of the original sequence, while the other possesses non-negativity. Another result establishes that an approximately monotone(increasing) sequence can be closely approximated by a non-decreasing sequence. A similar assertion can be made for approximately convex sequence. A sequence $\big<u_n\big>_{n=0}^{\infty}$ is said to be approximately convex (or $\varepsilon$-convex) if the following inequality holds under the mentioned assumptions \begin{equation*} u_{i}- u_{i-1}\leq u_{j}-u_{j-1}+\varepsilon\quad\quad\mbox{ where}\quad\quad i,j\in\mathbb{N} \quad \mbox{with} \quad i<j. \end{equation*} We proved that an approximately convex sequence can be written as the algebraic difference of two specific types of sequences. The initial sequence contains sequential convexity property, while the other sequence possesses the Lipschitz property. Moreover, we introduce an operator $\mathbb{T}$, that termed as a twisting operator. In a compact interval $I(\subseteq\mathbb{R})$; we characterized the convex function with this newly introduced operator. Besides various sequence decomposition results and study related to $\mathbb{T}$-operator on convex function; a characterization regarding non-negative sequential convexity, a fractional inequality, implication of $\mathbb{T}$ operator on various types of functions, relationship between a convex function and a convex sequence are also included.