log convex function

let f:(0,infty) -> (0,infty) be convex and increasing with

f(x)/x -> infty as x -> infty.


We say f is log convex if g(x):= log(f(x)) is convex.

Now suppose we only have (A):

f'(x)/ f(x) <= 2 f'(y) / f(y) for 0<x<y< infty.

((If the 2 was a 1 then g' would be increasing and hence f would be log convex.))

QUESTION: Is there examples of functions f which satisfy the above (so f increasing convex, superlinear at infty and satisfies (A))

which are NOT log convex??


thanks
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