Generalization of Subadditive, Monotone and Convex Functions

TL;DR

This paper generalizes classical subadditivity to higher orders, introduces approximate and periodic monotonicity concepts, and explores star convexity, providing new insights into the structural properties of these functions.

Contribution

It extends subadditivity to any order, introduces approximate subadditivity and periodic monotonicity, and investigates star convexity and its relation to star convex bodies.

Findings

Higher-order subadditivity implies ordinary subadditivity of roots.

Approximately subadditive functions can be decomposed into subadditive and bounded parts.

Periodically monotone functions can be decomposed into monotone and periodic functions.

Abstract

Let $I\subseteq{\mathbb{R_+}}$ be a non empty and non singleton interval where ${\mathbb{R_+}}$ denotes the set of all non negative numbers. A function $\Phi: I\to {\mathbb{R_+}}$ is said to be subadditive if for any $x,y$ and $x+y\in I$, it satisfies the following inequality $$\Phi(x+y)\leq \Phi(x)+\Phi(y).$$ In this paper, we consider this ordinary notion of subadditivity is of order $1$ and generalized the concept for any order $n$, where $n\in{\mathbb{N}}$. We establish that $n^{th}$ square root of a $n^{th}$ order subadditive function possesses ordinary subadditivity. We also introduce the notion of approximately subadditive function and showed that it can be decomposed as the algebraic summation of a subadditive and a bounded function. Another important newly introduced concept is Periodical monotonicity. A function $f:I\to{\mathbb{R}}$ is said to be periodically monotone with a period $d>0$ if the following holds $$ f(x)\leq f(y)\qquad\mbox{for all}\quad x,y\in I\qquad{with}\quad y-x\geq d. $$ One of the obtained results is that under a minimal assumption on $f$; this type of function can be decomposed as the sum of a monotone and a periodic function whose period is $d$. Towards the end of the paper, we discuss about star convexity. A function $f: I\to{\mathbb{R}}$ is said to be star-convex if there exists a point $p\in I$ such that for any $x\in I$ and for all $t\in [0,1]$; it satisfies either one of the following conditions. $$ t(x,f(x)) +(1-t)(p,f(p))\in epi(f) \quad \mbox{or} \quad hypo(f). $$ We studied the structural properties and showed relationship of it with star convex bodies.