Log of Gamma Function is Convex on Positive Reals/Proof 3

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 use the Euler Form of the Gamma function and directly calculate the second derivative of $\ln \Gamma \left({z}\right)$.

The limit interchange is justified because Gamma Function is Smooth on Positive Reals.

Logarithmic convexity then follows from Second Derivative of Strictly Convex Real Function is Strictly Positive.