Q-subdifferential of Jensen-convex functions

Zoltán Boros, Zsolt Páles

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

A real function is called radially Q-differentiable at the point x if, for every real number h, the finite limit dQ f ( x, h ) of the ratio ( f ( x + r h ) - f ( x ) ) / r exists whenever r tends to zero through the positive rationals. We establish that, in particular, Jensen-convex functions are everywhere radially Q-differentiable. Moreover, if f is Jensen-convex, then, for each x, the mapping h {mapping} dQ f ( x, h ) is subadditive, and it is an upper bound for any additive mapping A satisfying the inequality f ( x ) + A ( y - x ) {less-than or slanted equal to} f ( y ) for every y. We also characterize all set-valued mappings built up from additive solutions A of this inequality with some Jensen-convex function f.

Original languageEnglish
Pages (from-to)99-113
Number of pages15
JournalJournal of Mathematical Analysis and Applications
Volume321
Issue number1
DOIs
Publication statusPublished - Sep 1 2006

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Keywords

  • Convex function
  • Cyclically monotone mapping
  • Differentiability
  • Subadditive function
  • Subdifferential
  • Subgradient

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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