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

# 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.
Keywords: inequalities, modulars, Orlicz-Musielak spaces, convexity, isotonicity, antiprojections.
Mathematics Subject Classification: 39B62, 41A65, 46E30.
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

Download this article's citation as:
a .bib file (BibTeX), a .ris file (RefMan), a .enw file (EndNote)
or export to RefWorks.

ISSN 1232−9274, e-ISSN 2300−6919, DOI https://doi.org/10.7494/OpMath
Copyright © 2003−2017 OPUSCULA MATHEMATICA
Contact: opuscula@agh.edu.pl
Made by Tomasz Zabawa

In accordance with EU legislation we advise you this website uses cookies to allow us to see how the site is used. All data is anonymized.
All recent versions of popular browsers give users a level of control over cookies. Users can set their browsers to accept or reject all, or certain, cookies.