A characterization of convex φ-functions

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.