Does complete monotonicity of f imply log-concavity of f? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T02:15:01Z http://mathoverflow.net/feeds/question/86646 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86646/does-complete-monotonicity-of-f-imply-log-concavity-of-f Does complete monotonicity of f imply log-concavity of f? J M del Castillo 2012-01-25T18:30:23Z 2012-01-26T13:15:19Z <p>Let f be a completely monotonic function with $f(0)=1$, that is, $f:[0, \infty) \rightarrow (0,1]$. My question is:</p> <p>Is f log concave, that is, is $(logf)''&lt;0$ or equivalently $f f''&lt; f'^2$. ?</p> <p>And what hapens if $f(0)=\infty$, that is if the function is: $f:(0, \infty) \rightarrow (0,\infty)$.</p> http://mathoverflow.net/questions/86646/does-complete-monotonicity-of-f-imply-log-concavity-of-f/86650#86650 Answer by Mark Meckes for Does complete monotonicity of f imply log-concavity of f? Mark Meckes 2012-01-25T19:00:49Z 2012-01-25T19:00:49Z <p>A counterexample in the second case is $f(x) = e^{1/x}$. A counterexample in the first case is then $f(x) = e^{1/(x+1) - 1}$.</p> http://mathoverflow.net/questions/86646/does-complete-monotonicity-of-f-imply-log-concavity-of-f/86659#86659 Answer by S. Sra for Does complete monotonicity of f imply log-concavity of f? S. Sra 2012-01-25T21:39:28Z 2012-01-25T21:44:39Z <p>Exercise 6 of <a href="http://books.google.de/books?id=xpT_Kl9OS24C&amp;lpg=PA68&amp;ots=XSAFMTu6X7&amp;dq=completely%2520monotonic%2520log%2520convex&amp;pg=PA68#v=onepage&amp;q=completely%2520monotonic%2520log%2520convex&amp;f=false" rel="nofollow">this book</a> shows that if $f: (0,\infty) \to \mathbb{R}$ is completely monotonic, then it must be log-convex. Hence, your second claim holds, with concavity replaced by convexity.</p> http://mathoverflow.net/questions/86646/does-complete-monotonicity-of-f-imply-log-concavity-of-f/86717#86717 Answer by Jose M Del Castillo for Does complete monotonicity of f imply log-concavity of f? Jose M Del Castillo 2012-01-26T13:15:19Z 2012-01-26T13:15:19Z <p>Actually, I meant log-convex instead of log-concave. I missed a minus sign in my derivations and this led to the conjecture that log(f) should be concave. I have found a more complete answer in this paper:</p> <p>Inequalities for Real Powers of Completely Monotonic Functions H. van Haeringen JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS Volume 210, Issue 1, 1 June 1997, Pages 102–113</p> <p>Theorem 1 establishes a series of inequalities for the derivatives of c.m. functions. In particular, by taking n=0 and m=1 in 3.2 we get the log-convexity of f. I suggest to read this paper because of the relrevance of Theorem 1.</p>