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Convexity with respect to families of means

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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 pth 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.

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Correspondence to Gyula Maksa.

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Dedicated to the 90th birthday of Professor János Aczél

The research was realized in the frames of the projects “National Excellence Program—Elaborating and operating an inland student and researcher personal support system” (Grant No.: TÁMOP 4.2.4. A/2-11-1-2012-0001) and “Supercomputer, the national virtual lab” (Grant No.: TÁMOP-4.2.2.C-11/1/KONV-2012-0010). These two projects were subsidized by the European Union and by the European Social Fund. The research was also supported by the Hungarian Scientific Research Fund (OTKA) (Grant No.: NK 81402).

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Maksa, G., Páles, Z. Convexity with respect to families of means. Aequat. Math. 89, 161–167 (2015).

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