MathDB

Convex function, two non-decreasing sequences

Source: ILL 1979 - Problem 61.

June 5, 2011

functioninequalities proposedinequalities

Problem Statement

There are two non-decreasing sequences

\{a_i\}

and

\{b_i\}

of

n

real numbers each, such that

a_i\le a_{i+1}

for each

1\le i\le n-1

, and

b_i\le b_{i+1}

for each

1\le i\le n-1

, and

\sum_{k=1}^{m}{a_k}\ge \sum_{k=1}^{m}{b_k}

where

m\le n

with equality for

m=n

. For a convex function

f

defined on the real numbers, prove that

\sum_{k=1}^{n}{f(a_k)}\le \sum_{k=1}^{n}{f(b_k)}

.

Back to Problems View on AoPS