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


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.


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

Mathematics Subject Classification

Primary: 33B15 26B25 39B22 


  1. 1.
    Himmel, M., Matkowski, J.: Homogeneous beta-type functions. J. Class. Anal. 10(1), 59–66 (2017)Google Scholar
  2. 2.
    Krull, W.: Bemerkungen zur Differenzengleichung \( g\left( x+1\right) -g\left( x\right) =F\left( x\right).\) I. Math. Nachr. 1, 365–376 (1948)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Krull, W.: Bemerkungen zur Differenzengleichung \( g\left( x+1\right) -g\left( x\right) =F\left( x\right).\) II. Math. Nachr. 2, 251–262 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Kuczma, M.: Functional equations in a single variable, Monografie Matematyczne, vol. 46, Polish Scientific Publishers, Warszawa (1968)Google Scholar
  5. 5.
    Kuczma, M.: An Introduction to the Theory of Functional Equations and Inequalities. Uniwersytet Ślaski, Pań stwowe Wydawnictwo Naukowe, Warszawa-Kraków (1985)zbMATHGoogle Scholar
  6. 6.
    Ng, C.T.: Functions generating Schur-convex sums, General Inequalities 5 (Oberwolfach, 1986), pp. 433–438, Internat. Schriftenreihe Numer. Math. 80, Birkhäuser, Basel (1987)Google Scholar

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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, 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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