Opuscula Math. 32, no. 1 (2012), 171-178
http://dx.doi.org/10.7494/OpMath.2012.32.1.171

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.

Full text (pdf)

• Bartosz Micherda
• University of Bielsko-Biała, Department of Mathematics and Computer Science, ul. Willowa 2, 43-309 Bielsko-Biała, Poland