Convexity with respect to families of means

In this paper we investigate continuity properties of functions $f:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ that satisfy the $(p,q)$-Jensen convexity inequality $$f(H_p(x,y))\le H_q(f(x),f(y))(x,y>0),$$ where $H_p$ stands for the $p$th power (or H"older) mean. One of the main results shows that there exist discontinuous multiplicative functions that are $(p,p)$-Jensen convex for all positive rational number $p$. A counterpart of this result states that if $f$ is $(p,p)$-Jensen convex for all $p\in P\subseteq$\mathbb{R}$_+$,where$P$isasetofpositiveLebesguemeasure,then$f$ must be continuous.