To the theorem on differentiation of inequalities between convex functions

Abstract

The following statement is proved in the article: For any constant $K > 1$, there exist upper unbounded convex functions $f$ and $g$ on $[0;+\infty)$ such that the inequality $f(x) \le g(x)$ holds for all $x \ge 0$, and for the set $$ E_1=\{ x \ge 0\,:\, f_{+}^{'}(x)\le K g_{+}^{'}(x) \} $$ holds $$ dE_{1} =\varliminf\limits_{R\to +\infty}\frac1{R}\text{\rm meas }(E_1\cap[0,R])=\frac {K-1}{K}, $$ where $f_{+}^{'}$ and $g_{+}^{'}$ are the right-hand derivatives of the functions $f$ and $g$, respectively.