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

Let $J$ denote the interval either $(0,1]$ or $[1,\mathrm{\infty })$. A positive function $f$ on $J$ with $f(1)=1$ is reffered to as a Weierstrass function if it fulfils the double inequality for $x,y\in J$: $$f(x)+f(y)-1\le f(xy)\le f(x)f(y)\mathrm{.}$$ 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 $f(x)=\frac{\ln (1+x)}{\ln 2}$, we obtain a new multiplicative inequality which, in turn, generalizes the classical Weierstrass inequality.