Is for a convex, continuous function f satisfying $Cf(s)f(t) \leq f(st) \leq Df(s)f(t)$ for some constants C>0 and D>0, implies that $f(t)=t^p$?

Question

If $f(t)=t^p$, then f satisfies $Df(s)f(t) \leq f(st) \leq Df(s)f(t)$ for all s,t>0, for some constants C=1>0 and D=1>0. Can we get any other example other than $t^p$.