Logarithm of Beta Function is Convex on Positive Reals

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

Let $\map \Beta {x, y}$ be the Beta function:
 * $\ds \map \Beta {x, y} := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \paren {1 - t}^{y - 1} \rd t$

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

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