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.
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Himmel, M., Matkowski, J. Beta-type functions and the harmonic mean. Aequat. Math. 91, 1041–1053 (2017). https://doi.org/10.1007/s00010-017-0498-3
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DOI: https://doi.org/10.1007/s00010-017-0498-3
Keywords
- Beta function
- Beta-type function
- Mean
- Harmonic mean
- Convex function
- Wright convex function
- Functional equation