Lemma 2

Let $f$ be a finite real function, $> 0$, defined and continuous on the product $I \times J$ of two open intervals of $\mathbf{R}$, and such that, for every $t \in J$ the function $x \mapsto f(x, t)$ is logarithmically convex and twice differentiable on $I$. Under these hypotheses, if the integral $g(x) = \int_J f(x, t) dt$ converges for every $x \in I$, then $g$ is logarithmically convex on $I$.

First we show that for every compact interval $K \subset J$ the function $g_K(x) = \int_K f(x, t) dt$ is logarithmically convex. Indeed, if $K = [a, b]$, the sequence of functions

converges simply to $g_K(x)$ on $I$ (II, p. 57, prop. 5) hence $\log g_n$ converges simply to $\log g_K$; by lemma 1 of VII, p. 310, $\log g_n$ is convex on $I$, so (I, p. 27, prop. 4) it is the same for $\log g_K$.

On the other hand, $g$ is the pointwise limit of the $g_K$ along the directed set of compact subintervals of $I$ (II, p. 64), so $\log g$ is the pointwise limit of the $\log g_K$; these last functions being convex on $I$, so is $\log g$ (I, p. 27, prop. 4).

One can show easily that lemmas 1 and 2 remain valid even when one does not assume that the functions are twice differentiable (VII, p. 327, exerc. 5).