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Euler's Beta function diagonalized and a related functional equation

Choczewski B., Wach-Michalik A.

Opuscula Mathematica

2004

Vol. 24, no. 1

35-41

Euler's Gamma function is the unique logarithmically convex solution of the functional equation (1), cf. the Proposition. In this paper we deal with the function beta: R+ → R+, beta(x) := B(x, x), where B(x, y) is the Euler Beta function. We prove that, whenever a function h is asymptotically comparable at the origin with the function a log +b, a > 0, if varphi: R+ → R+ satisfies equation (5) and the function h o varphi is continuous and ultimately convex, then varphi = beta.