Opuscula Math. 32, no. 1 (2012), 171-178

Opuscula Mathematica

A characterization of convex φ-functions

Bartosz Micherda

Abstract. The properties of four elements \((LPFE)\) and \((UPFE)\), introduced by Isac and Persson, have been recently examined in Hilbert spaces, \(L^p\)-spaces and modular spaces. In this paper we prove a new theorem showing that a modular of form \(\rho_{\Phi}(f)=\int_{\Omega}\Phi(t,|f(t)|)d\mu(t)\) satisfies both \((LPFE)\) and \((UPFE)\) if and only if \(\Phi\) is convex with respect to its second variable. A connection of this result with the study of projections and antiprojections onto latticially closed subsets of the modular space \(L^{\Phi}\) is also discussed.

Keywords: inequalities, modulars, Orlicz-Musielak spaces, convexity, isotonicity, antiprojections.

Mathematics Subject Classification: 39B62, 41A65, 46E30.

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  • Bartosz Micherda
  • University of Bielsko-Biała, Department of Mathematics and Computer Science, ul. Willowa 2, 43-309 Bielsko-Biała, Poland
  • Received: 2010-12-10.
  • Accepted: 2011-06-11.
Opuscula Mathematica - cover

Cite this article as:
Bartosz Micherda, A characterization of convex φ-functions, Opuscula Math. 32, no. 1 (2012), 171-178, http://dx.doi.org/10.7494/OpMath.2012.32.1.171

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