An Integral Jensen Inequality For Convex Multifunctions |
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Authors: | Janusz Matkowski Kazimierz Nikodem |
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Institution: | 1. Department of Mathematics, Technical University, Willowa 2, 43-300, Bielsko-Bia?a, Poland
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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 ? : Ω ? D such that conv?(Ω) ? D, $$\int_{\Omega}(F\ o\ \phi)d\mu \subset F\Bigg(\int_{\Omega}\phi d\mu\Bigg).$$ |
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