Eulers Beta function diagonalized and a related functional equation

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 $\beta: \mathbb{R}_{+} \rarr \mathbb{R}_{+}, \beta(x) := B(x, x)$, 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 $\varphi: \mathbb{R}_{+} \rarr \mathbb{R}_{+}$ satisfies equation (5) and the function $h \circ \varphi$ is continuous and ultimately convex, then $\varphi = \beta$.