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Aequationes mathematicae

, Volume 91, Issue 6, pp 1041–1053 | Cite as

Beta-type functions and the harmonic mean

  • Martin HimmelEmail author
  • Janusz Matkowski
Open Access
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Abstract

For arbitrary \(f:\left( a,\infty \right) \rightarrow \left( 0,\infty \right) ,\) \(a\ge 0,\) the bivariable function \(B_{f}:\left( a,\infty \right) ^{2}\rightarrow \left( 0,\infty \right) ,\) related to the Euler Beta function, is considered. It is proved that \(B_{f\text { }}\)is a mean iff it is the harmonic mean H. Some applications to the theory of iterative functional equations are given.

Keywords

Beta function Beta-type function Mean Harmonic mean Convex function Wright convex function Functional equation 

Mathematics Subject Classification

Primary: 33B15 26B25 39B22 

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Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Faculty of Mathematics, Computer Science and EconometricsUniversity of Zielona GóraZielona GóraPoland

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