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ln b_i is convex if b_0,c b_1, c^2b_2,c^3b_3,... is convex for all c > 0

Source: 1978 Swedish Mathematical Competition p4

March 26, 2021
algebraconvexlogarithmSequence

Problem Statement

b0,b1,b2,…b_0, b_1, b_2, \dots is a sequence of positive reals such that the sequence b0,cb1,c2b2,c3b3,…b_0,c b_1, c^2b_2,c^3b_3,\dots is convex for all c>0c > 0. (A sequence is convex if each term is at most the arithmetic mean of its two neighbors.) Show that ln⁡b0,ln⁡b1,ln⁡b2,…\ln b_0, \ln b_1, \ln b_2, \dots is convex.
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