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arXiv · 1607.04080

Reducible means and reducible inequalities

Tibor Kiss·Zsolt Páles
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Abstract

It is well-known that if a real valued function acting on a convex set satisfies the $n$-variable Jensen inequality, for some natural number $n\geq 2$, then, for all $k\in\{1,\dots, n\}$, it fulfills the $k$-variable Jensen inequality as well. In other words, the arithmetic mean and the Jensen inequality (as a convexity property) are both reducible. Motivated by this phenomenon, we investigate this property concerning more general means and convexity notions. We introduce a wide class of means which generalize the well-known means for arbitrary linear spaces and enjoy a so-called reducibility property. Finally, we give a sufficient condition for the reducibility of the $(M,N)$-convexity property of functions and also for Hölder--Minkowski type inequalities.

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Tibor Kiss, Zsolt Páles. 2016-07-14. Reducible means and reducible inequalities. https://doi.org/10.1007/s00010-016-0459-2

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