What does log convexity mean? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T06:13:48Z http://mathoverflow.net/feeds/question/51054 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/51054/what-does-log-convexity-mean What does log convexity mean? Stopple 2011-01-03T20:04:48Z 2011-06-16T16:15:24Z <p>The Bohrâ€“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.</p> <p>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. </p> <p>The wikipedia article <a href="http://en.wikipedia.org/wiki/Logarithmic_convexity" rel="nofollow">http://en.wikipedia.org/wiki/Logarithmic_convexity</a> 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 $\Gamma(x)$.</p> <p>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 </p> <p><a href="http://mathoverflow.net/questions/23229/importance-of-log-convexity-of-the-gamma-function" rel="nofollow">http://mathoverflow.net/questions/23229/importance-of-log-convexity-of-the-gamma-function</a></p> <p>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.</p> <p>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.</p> <p>Any insight would be welcome.</p> http://mathoverflow.net/questions/51054/what-does-log-convexity-mean/51060#51060 Answer by Igor Rivin for What does log convexity mean? Igor Rivin 2011-01-03T20:42:59Z 2011-01-03T20:42:59Z <p>Log convexity pops up in lots of different places (eg, the determinant of positive definite matrices is log-concave, the perron-frobenius eigenvalue of positive matrices is log convex, volumes of convex bodies are log concave, etc, etc). For a discussion having absolutely nothing to do with the gamma function, see S. Boyd and L. Vanderberghe's book "Convex optimization" (page 105). PDF available for free on Boyd's web site (google "Stephen Boyd Stanford")</p> http://mathoverflow.net/questions/51054/what-does-log-convexity-mean/51061#51061 Answer by Alex R. for What does log convexity mean? Alex R. 2011-01-03T20:49:40Z 2011-01-03T20:49:40Z <p>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</p> http://mathoverflow.net/questions/51054/what-does-log-convexity-mean/51073#51073 Answer by John D. Cook for What does log convexity mean? John D. Cook 2011-01-03T23:03:19Z 2011-01-03T23:03:19Z <p>Many common probability densities are log-concave, which turns out to be a very useful property. For one thing, it's a way to quantify that a density is something like a normal density, perhaps sufficiently like a normal for a theorem to generalize. </p> <p>Also, og-concave densities satisfy nice theorems. For example, if the PDF of a random variable is log-concave, the CDF is as well. Also, the convolution of two log-concave densities is also log-concave.</p> http://mathoverflow.net/questions/51054/what-does-log-convexity-mean/51097#51097 Answer by S. Sra for What does log convexity mean? S. Sra 2011-01-04T09:16:03Z 2011-06-16T16:15:24Z <p>In my opinion, some of the reasons that make log-convexity (l.c.) nice / useful are:</p> <ol> <li><p>If $f$ and $g$ are l.c., then $f+g$ and $fg$ are both log-convex (note log-concavity on the other hand is not closed under addition, though as Allen commented above, it is closed under convolution (Prekopá's theorem))</p></li> <li><p>L.c. functions are related to <em>completely monotonic</em> functions (i.e., functions for which $(-1)^n f^{(n)}(x) \ge 0$ for all $x > 0$, and $n \in \mathbb{N}$), in that any completely monotonic function is also l.c.; If I recall correctly, one place completely monotonic functions come up naturally is when considering Laplace transforms. Another example: the Laplace transform of any non-negative function is l.c.</p></li> <li><p>L.c. is closely related to multiplicatively convex functions (i.e., $f(\sqrt{xy}) \le \sqrt{f(x)f(y)}$), and it is known that the logarithmic integral is multiplicatively convex. </p> <p>Many more nice properties may be found in the book: Convex functions and their applications: A contemporary approach by C. P. Niculescu and L.-E. Persson.</p></li> </ol>