MathDB

33

Part of 1978 IMO Longlists

Problems(1)

Convex functions and logarithms

Source:

10/30/2010
A sequence (an)0∞(a_n)^{\infty}_0 of real numbers is called convex if 2an≤an−1+an+12a_n\le a_{n-1}+a_{n+1} for all positive integers nn. Let (bn)0∞(b_n)^{\infty}_0 be a sequence of positive numbers and assume that the sequence (αnbn)0∞(\alpha^nb_n)^{\infty}_0 is convex for any choice of α>0\alpha > 0. Prove that the sequence (log⁡bn)0∞(\log b_n)^{\infty}_0 is convex.
functionlogarithmsinequalitiesalgebra proposedalgebra