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) $.