Lemma 1

Let $f$ and $g$ be two functions $> 0$ and twice differentiable on an open interval $I$. If $f$ and $g$ are logarithmically convex functions on $I$, then $f + g$ is logarithmically convex on $I$.

The relation $D^2 (\log f(x)) \geq 0$ can be written $f(x)f''(x) - (f'(x))^2 \geq 0$. We are reduced to showing that the relations $a > 0, a' > 0, ac - b^2 \geq 0, a'c' - {b'}^2 \geq 0$ imply $(a + a')(c + c') - (b + b')^2 \geq 0$; now, when $a > 0$, the relation $ac - b^2 \geq 0$ is equivalent to the fact that the quadratic form $ax^2 + 2bxy + cy^2$ is $\geq 0$ on $\mathbf{R}^2$, and it is clear that if

on $\mathbf{R}^2$ then also $(a + a')x^2 + 2(b + b')xy + (c + c')y^2 \geq 0$ on $\mathbf{R}^2$.