Continuous solutions of the equation \(x+g(y+f(x))=y+g(x+f(y))\)


The equation \(x+g(y+f(x))=y+g(x+f(y))\) was introduced by Marcin E. Kuczma in connection with his research on compatible means. Kuczma determined the analytic solutions of the equation in order to prove that compatible homogeneous analytic means are necessarily power means. Kuczma’s result was improved by J. Sikorska, who determined the twice differentiable solutions, and then by N. Brillouët-Belluot, who found all differentiable solutions. In this paper we determine all continuous solutions. As a corollary we find that compatible continuous homogeneous means are necessarily power means.

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Funding was provided by Hungarian National Foundation for Scientific Research (Grant No. K124749).

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Correspondence to Miklós Laczkovich.

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Laczkovich, M. Continuous solutions of the equation \(x+g(y+f(x))=y+g(x+f(y))\). Aequat. Math. 93, 1139–1157 (2019).

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Mathematics Subject Classification

  • 39B22