2023/07/02 by Goswami, Angshuman R. · 1 citation
#FOS: Mathematics #General Mathematics (math.GM)
paper · doi:10.48550/arxiv.2308.00704
Let I⊆\mathbbR+ be a non empty and non singleton interval where \mathbbR+ denotes the set of all non negative numbers. A function Φ: I→ \mathbbR+ is said to be subadditive if for any x,y and x+y∈ I, it satisfies the following inequality Φ(x+y)≤ Φ(x)+Φ(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∈ℕ. We establish that nth square root of a nth 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→ℝ is said to be periodically monotone with a period d>0 if the following holds f(x)≤ f(y)\qquadfor all x,y∈ I with y-x≥ 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→ℝ is said to be star-convex if there exists a point p∈ I such that for any x∈ I and for all t∈ [0,1]; it satisfies either one of the following conditions. t(x,f(x)) +(1-t)(p,f(p))∈ epi(f) or hypo(f). We studied the structural properties and showed relationship of it with star convex bodies.