Logarithm of Beta Function is Convex on Positive Reals

Theorem
Let $x, y \in \R$ be real numbers.

Let $\Beta \left({x, y}\right)$ be the Beta function:
 * $\displaystyle \Beta \left({x, y}\right) := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \left({1 - t}\right)^{y - 1} \ \mathrm d t$

Let $y \in \R_{>0}$ be given.

Then $\ln \left({\Beta \left({x, y}\right)}\right)$ is a convex function of $x$ on $\R_{>0}$.