Generalization of Subadditive, Monotone and Convex 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.