A CHARACTERIZATION OF CONVEX phi-FUNCTIONS

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) = integral(Omega) Phi(t, vertical bar f(t)vertical bar) 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.