A new characterization of convex φ-functions with a parameter

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.