Opuscula Math. 35, no. 2 (2015), 191-197
http://dx.doi.org/10.7494/OpMath.2015.35.2.191

Opuscula Mathematica

# A new characterization of convex φ-functions with a parameter

Bartosz Micherda

Abstract. We show that, under some additional assumptions, all projection operators onto latticially closed subsets of the Orlicz-Musielak space generated by $$\Phi$$ are isotonic if and only if $$\Phi$$ is convex with respect to its second variable. A dual result of this type is also proven for antiprojections. This gives the positive answer to the problem presented in Opuscula Mathematica in 2012.

Keywords: Orlicz-Musielak space, convex function, isotonic operator, projection operator, antiprojection operator.

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

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• Bartosz Micherda
• University of Bielsko-Biała, Department of Mathematics, ul. Willowa 2, 43-309 Bielsko-Biała, Poland
• Communicated by Henryk Hudzik.
• Accepted: 2014-06-18.
• Published online: 2014-11-18.