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

Theorem
Let $\Gamma: \R_{>0} \to \R$ be the Gamma function, restricted to the strictly positive real numbers.

Let $\ln$ denote the natural logarithm function.

Then the composite mapping $\ln \circ \operatorname \Gamma$ is a convex function.

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

Let $0 < \delta < \Delta$.

Then:

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

The result follows by definition of convex function.