Results in Mathematics

, Volume 26, Issue 3–4, pp 348–353 | Cite as

An Integral Jensen Inequality For Convex Multifunctions

  • Janusz Matkowski
  • Kazimierz Nikodem
Article
First Online:
Received:

Abstract

We prove the following multivalued version of the Jensen integral inequality. Let X, Y be Banach spaces and D ⊂ X an open and convex set. If F: D ↦ cl(Y) is a continuous convex function, then for each normalized measure space (Ω, S, μ), and for all μ-integrable functions ϕ : Ω ↦ D such that convϕ(Ω) ⊂ D,
$$\int_{\Omega}(F\ o\ \phi)d\mu \subset F\Bigg(\int_{\Omega}\phi d\mu\Bigg).$$

1991 Mathematics Subject Classification

26B25 26E25 26A51 

Key words and phrases

convex functions multivalued functions integral Jensen inequality sub differential 
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Copyright information

© Birkhäuser Verlag, Basel 1994

Authors and Affiliations

  • Janusz Matkowski
    • 1
  • Kazimierz Nikodem
    • 1
  1. 1.Department of MathematicsTechnical UniversityBielsko-BiałaPoland

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