IMO 1978 LL SWE33

Origin: SWE

Problem

A sequence (an)\infty of real numbers is called convex if 2an \leq an−1 +an+1 for all positive integers n. Let (bn)\infty 0 be a sequence of positive numbers and assume that the sequence (\alphanbn)\infty 0 is convex for any choice of \alpha > 0. Prove that the sequence (log bn)\infty 0 is convex.