In this paper we investigate continuity properties of functions
$${f : \mathbb {R}_+ \to \mathbb {R}_+}$$
that satisfy the (*p*, *q*)-Jensen convexity inequality
$$f\big(H_p(x, y)\big) \leq H_q(f(x), f(y)) \qquad(x, y > 0),$$
where *H*_{p} stands for the *p*th power (or Hölder) mean. One of the main results shows that there exist discontinuous multiplicative functions that are (*p*, *p*)-Jensen convex for all positive rational numbers *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* is a set of positive Lebesgue measure, then *f* must be continuous.