A pair of linear functional inequalities and a characterization of $L^{p}$-norm

Tom 85 / 2005

Dorota Krassowska, Janusz Matkowski Annales Polonici Mathematici 85 (2005), 1-11 MSC: Primary 39B72, 26D15; Secondary 46E30. DOI: 10.4064/ap85-1-1

Streszczenie

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.

Autorzy

  • Dorota KrassowskaFaculty of Mathematics, Informatics and Econometry
    University of Zielona Góra
    PL-65-246 Zielona Góra, Poland
    e-mail
  • Janusz MatkowskiFaculty of Mathematics, Informatics and Econometry
    University of Zielona Góra
    PL-65-246 Zielona Góra, Poland
    and
    Institute of Mathematics
    Silesian University
    PL-40-007 Katowice, Poland
    e-mail

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