On Characterizations of Convex and Approximately Subadditive Sequences

Abstract

A sequence $\Big(u_n\Big)_{n=0}^{\infty}$ is said to be convex if it satisfies the following inequality $$ 2u_n\leq u_{n-1}+u_{n+1}\qquad \mbox{for all}\qquad n\in\mathbb{N}. $$ We present several characterizations of convex sequences and demonstrate that such sequences can be locally interpolated by quadratic polynomials. Furthermore, the converse assertion of this statement is also established. On the other hand, a sequence $\Big(u_n\Big)_{n=1}^{\infty}$ is called approximately subadditive if for a fixed $\epsilon>0$ and any partition $n_1,\cdots,n_k$ of $n\in\mathbb{N}$; the following discrete functional inequality holds true $$ u_n\leq u_{n_1}+\cdots+ u_{n_k}+\varepsilon. $$ We show Ulam's type stability result for such sequences. We prove that an approximately subadditive sequence can be expressed as the algebraic summation of an ordinary subadditive and a non-negative sequence bounded above by $\varepsilon.$ A proposition portraying the linkage between the convex and subadditive sequences under minimal assumption is also included.