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

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.