Does complete monotonicity of f imply log-concavity of f? - MathOverflow

Does complete monotonicity of f imply log-concavity of f?

2012-01-25T18:30:23Z

Let f be a completely monotonic function with $f(0)=1$, that is, $ f:[0, \infty) \rightarrow (0,1] $. My question is:

Is f log concave, that is, is $(logf)''<0$ or equivalently $ f f''< f'^2 $. ?

And what hapens if $f(0)=\infty$, that is if the function is: $ f:(0, \infty) \rightarrow (0,\infty) $.

Answer by Mark Meckes for Does complete monotonicity of f imply log-concavity of f?

2012-01-25T19:00:49Z

A counterexample in the second case is $f(x) = e^{1/x}$. A counterexample in the first case is then $f(x) = e^{1/(x+1) - 1}$.

Answer by S. Sra for Does complete monotonicity of f imply log-concavity of f?

2012-01-25T21:39:28Z

Exercise 6 of this book 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.

Answer by Jose M Del Castillo for Does complete monotonicity of f imply log-concavity of f?

2012-01-26T13:15:19Z

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:

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

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.