Weierstrass functions and a generalization of the additive-multiplicative Weierstrass inequality

Abstract

Let $J$ denote the interval either $(0,1]$ or $ [1, \infty)$. A positive function $f$ on $J$ with $f(1) =1$ is reffered to as a Weierstrass function if it fulfils the double inequality for $x,y \in J$: $$f(x) + f(y) -1 \leq f(xy) \leq f(x)f(y). $$ By means of such functions we can extend the classical Weierstrass inequality (the above inequality for $f(x) = x$) to some trigonometric, Euler gamma, and log functions. Utilizing the Weierstrass property of $f(x) = \frac{\ln (1+x)}{\ln2}$, we obtain a new multiplicative inequality which, in turn, generalizes the classical Weierstrass inequality.