Gauss-composition of means and the solution of the Matkowski--Sutô problem

By Z. Daróczy, Zsolt Páles. Publicationes mathematicae (Debrecen), 2002

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In this paper the so-called Matkowski–Sutô problem is completely solved, that is, continuous and strictly monotonic functions φ and ψ defined on an open real interval I are determined such that the functional equation (1) φ−1 φ(x) + φ(y) 2 + ψ−1 ψ(x) + ψ(y) 2 = x + y holds for all x, y ∈ I. The above equation belongs to the class of so-called composite functional equations that does not possess a regularity theory such as known for non-composite equations. The main results of the paper offer new methods to obtain higher-order regularity properties of the unknown functions φ and ψ. First, based on Lebesgue’s theorem on the almost everywhere differentiability of monotonic functions, the local Lipschitz property of φ and ψ and their inverses is shown. Then the differentiability of these functions is proved in a subinterval of I. Finally, using Baire’s theorem on the continuity properties of derivative functions, the continuous differentiability of φ and ψ in a subinterval is deduced. After these regularity properties, the equation is solved in the subinterval so obtained with earlier methods of the authors. The proof is then completed by using the extension theorem due to the authors and Gy. Maksa. The main result obtained is a generalization of that of Sutô (1914) and Matkowski (1999). As application, the connection to Gauss composition of means, the equality problem of quasi-arithmetic means and conjugate arithmetic means is discussed and solved. Mathematics Subject Classification: Primary 39B12, 39B22; Secondary 26A24, 26A21.

Subfields: Computational Mechanics, Geometry and Topology

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