ISL 1979 Problem 61

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