Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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) $.

share|improve this question

3 Answers 3

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

share|improve this answer
    
Log of both examples is like 1/x, which satisfies the negative derivative condition. What am I missing that makes them counterexamples? Gerhard "Ask Me About System Design" Paseman, 2012.01.25 –  Gerhard Paseman Jan 25 '12 at 19:36
    
Both functions are log-convex, not log-concave. –  Mark Meckes Jan 25 '12 at 20:01
    
So then when the original poster says (log f)' < 0, they mean f is log convex? Gerhard "Sometimes Confuses Up And Down" Paseman, 2012.01.25 –  Gerhard Paseman Jan 25 '12 at 21:07
    
The poster said (log f)''<0 (second derivative, not first). Although that's apparently the opposite of what was meant. –  Mark Meckes Jan 26 '12 at 15:10

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.

share|improve this answer

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.