Construction of Weights on the Semigroup $({\mathbb N}, \, +)$ Using some Standard Functions

Abstract

A \emph{weight} on the semigroup $(\mathbb N, +)$ of natural numbers is a function $\omega : {\mathbb N} \longrightarrow (0, \infty)$ satisfying the submultiplicativity $\omega(m + n) \leq \omega(m)\omega(n)$ for all $m, n \in {\mathbb N}$. In this simple paper, we exhibit that some standard functions such as $c\cosh(n)$, $c\sinh(n)$, $ n^{k}+c$, $(n + c)^{k}$, $e^{n^c}$, $e^{-n^c}$, $\log(n^k) + c$, $[\log(n) + c]^k$, and much more are weights on $\mathbb N$ under certain conditions on the constant $c$.