The Bohr–Mollerup theorem characterizes the Gamma function $\Gamma(x)$ as the unique function $f(x)$ on the positive reals such that $f(1)=1$, $f(x+1)=xf(x)$, and $f$ is logarithmically convex, i.e. $\log(f(x))$ is a convex function.
What meaning or insight do we draw from log convexity? There's two obvious but less than helpful answers. One is that log convexity means exactly what the definition says, no more and no less. The other is the more or less circular one that since the Gamma function is so important, any property that characterizes it is also significant.
The wikipedia article http://en.wikipedia.org/wiki/Logarithmic_convexity
point out that "a logarithmically convex function is a convex function, but the converse is not always true" with the counterexample of $f(x)=x^2$. The only logarithmically convex examples in the article come trivially from exponentiating convex functions, and the example $\Gamma(x)$.
Let me say in advance that I'm less interested in the Gamma function than I am in the notion of log convexity, so this question is not a duplicate of
Importance of Log Convexity of the Gamma Function
A thoughtful answer by Andrey Rekalo to that question, is that functions which can be realized as finite of moments of Borel measures are log convex functions. But I'm more interested in things that are implied by (v. imply) log convexity.
My real motivation is the fact that the Riemann Hypothesis implies that the Hardy function is log convex for sufficiently large $t$. (The Hardy function $Z(t)$ is just $\zeta(1/2+it)$ with the phase taken out, so $Z(t)$ is real valued and $|Z(t)|=|\zeta(1/2+it)|$.) This is in Edward's book 'Riemann's Zeta Function' Section 8.3, in the language that RH $\Rightarrow Z^\prime/Z$ is monotonic. This says that between consecutive real zeros, $-\log|Z(t)|$ is convex.
Any insight would be welcome.
Best Answer
From the comments, log convexity leads one to conclude that Riemann Hypothesis implies Lindelof hypothesis. The implication of log convexity comes from Hadamard Three Circle Theorem