Weierstrass functions and a generalization of the additive-multiplicative Weierstrass inequality
Halina Wiśniewska
TL;DR
The paper develops a framework for Weierstrass functions on the domains $(0,1]$ and $[1,\infty)$, introducing left and right variants and posing questions about their existence beyond the identity. It provides a practical sufficiency criterion, via $H_f(x)=x f'(x)$ (and a $G_f$‑criterion for the $C^2$ case), for constructing Weierstrass functions and proves closure under products and compositions. It then demonstrates concrete instances using log, trigonometric, and Euler gamma–based functions, deriving generalized additive–multiplicative inequalities that extend the classical Weierstrass inequality. These results broaden the scope of Weierstrass-type inequalities and connect submultiplicativity with functional inequalities for important special functions.
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.