MathDB

61

Part of 1979 IMO Longlists

Problems(1)

Convex function, two non-decreasing sequences

Source: ILL 1979 - Problem 61.

6/5/2011
There are two non-decreasing sequences {ai}\{a_i\} and {bi}\{b_i\} of nn real numbers each, such that ai≤ai+1a_i\le a_{i+1} for each 1≤i≤n−11\le i\le n-1, and bi≤bi+1b_i\le b_{i+1} for each 1≤i≤n−11\le i\le n-1, and ∑k=1mak≥∑k=1mbk\sum_{k=1}^{m}{a_k}\ge \sum_{k=1}^{m}{b_k} where m≤nm\le n with equality for m=nm=n. For a convex function ff defined on the real numbers, prove that ∑k=1nf(ak)≤∑k=1nf(bk)\sum_{k=1}^{n}{f(a_k)}\le \sum_{k=1}^{n}{f(b_k)}.
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