A pair of linear functional inequalities and a characterization of $L^{p}$-norm
Volume 85 / 2005
Annales Polonici Mathematici 85 (2005), 1-11
MSC: Primary 39B72, 26D15; Secondary 46E30.
DOI: 10.4064/ap85-1-1
Abstract
It is shown that, under some general algebraic conditions on fixed real numbers $a,b,\alpha ,\beta $, every solution $f:\mathbb{R}\rightarrow \mathbb{R}$ of the system of functional inequalities $f(x+a)\leq f(x)+\alpha ,$ $f(x+b)\leq f(x)+\beta $ that is continuous at some point must be a linear function (up to an additive constant). Analogous results for three other similar simultaneous systems are presented. An application to a characterization of $L^{p}$-norm is given.
