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Reducible means and reducible inequalities

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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 \(n\ge 2\), 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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Authors and Affiliations

  1. Institute of Mathematics, University of Debrecen, Egyetem tér 1, Debrecen, 4032, Hungary

    Tibor Kiss & Zsolt Páles

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  1. Tibor Kiss
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  2. Zsolt Páles
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Correspondence to Zsolt Páles.

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The research of the first author has been supported through the New National Excellence Program of the Ministry of Human Capacities. The research of the second author has been supported by the Hungarian Scientific Research Fund (OTKA) Grant K111651.

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Kiss, T., Páles, Z. Reducible means and reducible inequalities. Aequat. Math. 91, 505–525 (2017). https://doi.org/10.1007/s00010-016-0459-2

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