Lemma 3

Let $\varphi$ be a continuous function and $> 0$ on an open interval $J$ contained in $[0, +\infty[$. If $I$ is an open interval such that the integral $g(x) = \int_J t^{x-1} \varphi(t) dt$ converges for all $x \in I$, then $g$ is logarithmically convex on $I$.

Indeed, $\log t^{x-1} = (x-1) \log t$ is a function of $x$ which is convex and twice differentiable for all $t > 0$, so lemma 2 applies.