Log of Gamma Function is Convex on Positive Reals/Proof 2

Proof
The strategy is to show that:
 * $\map \ln {\map \Gamma {\dfrac x 2 + \dfrac y 2} } \le \dfrac 1 2 \map \ln {\map \Gamma x} + \dfrac 1 2 \map \ln {\map \Gamma y}$

Let $0 < \delta < \Delta$.

Then:

Letting $\delta \to 0$ and $\Delta \to \infty$, $(1)$ becomes equivalent to:
 * $\paren {\map \Gamma {\dfrac {x + y} 2} }^2 \le \paren {\map \Gamma x} \paren {\map \Gamma y}$

The result follows by definition of convex function.