ILL 1978 Problem 33

A sequence $(a_n)^{\infty}_0$ of real numbers is called convex if $2a_n\le a_{n-1}+a_{n+1}$ for all positive integers $n$ . Let $(b_n)^{\infty}_0$ be a sequence of positive numbers and assume that the sequence $(\alpha^nb_n)^{\infty}_0$ is convex for any choice of $\alpha > 0$ . Prove that the sequence $(\log b_n)^{\infty}_0$ is convex.