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Acta Mathematica Hungarica

, Volume 143, Issue 1, pp 220–231 | Cite as

Invariant Translative Mappings and a Functional Equation

  • H. IzumiEmail author
  • J. Matkowski
Article
  • 115 Downloads

Abstract

Let \({K,M,N : \mathbb{R}^{2} \rightarrow \mathbb{R}}\) be translative functions. Then K is invariant with respect to the mapping \({(M,N) : \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}}\) if and only if the functions \({h = K(\cdot, 0), f = M(\cdot, 0), g = N(\cdot, 0)}\) satisfy the functional equation
$$h(x) = h(f(x) - g(x)) + g(x),\,\, x\in \mathbb{R}.$$
If K, M, N are means, then h(0) =  f(0) =  g(0) = 0. The formal power solutions and analytic solutions of this functional equation, satisfying these initial conditions, are considered.

Key words and phrases

mean invariant mean formal powers series analytic function entire function functional equation 

Mathematics Subject Classification

primary 30D05 secondary 26E60 39B22 39B32 

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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2014

Authors and Affiliations

  1. 1.Chiba Institute of TechnologyNarashinoJapan
  2. 2.Faculty of Mathematics, Computer Science and EconometricsUniversity of Zielona GóraZielona GóraPoland

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