Linear functional inequalities—a general theory and new special cases
Volume 438 / 2006
Abstract
This paper solves the functional inequality% $$ af(s)+bf(t)\geq f(\alpha s+\beta t),\ \quad s,t>0, $$ with four positive parameters $a,b,\alpha,\beta$ arbitrarily fixed. The unknown function $f:(0,\infty )\rightarrow {\sym R}$ is assumed to satisfy the regularity condition% $$ \limsup_{s\rightarrow 0+}f(s)\leq 0. $$ The paper partitions the space of parameters into regions where the inequality has qualitatively similar classes of solutions, estimates the rate of growth of the solutions, determines their signs, and identifies all the parameters such that the solutions form small nontrivial classes of functions. In addition to the well known cases of convex and subadditive functions, examples of such classes of functions include nonnegative power functions $( 0,\infty )\ni t\mapsto f(1) t^{p}$ for fixed $p\geq 1$, nonpositive power functions $( 0,\infty ) \ni t\mapsto f(1) t^{p}$ for fixed $p\in (0,1]$, and convex functions satisfying some homogeneity conditions.
