# Sequence $a$ majorizes sequence $b$ and sequence $c$ is decreasing. Show that $\sum_i a_i c_i \geq \sum_i b_i c_i$.

The sequence $a_1,a_2,\ldots$ majorizes the sequence $b_1,b_2,\ldots$. Furthermore, the sequence $c_1,c_2\ldots$ is decreasing. Is there a simple known inequality to show that $\sum_i a_i c_i \geq \sum_i b_i c_i$?

We can assume $a,b,c$ are finite positive sequences.

• Is the sequence $\{c_i\}$ non-negative? – carmichael561 May 4 '17 at 0:18
• yes, I'll edit that. – HTV May 4 '17 at 0:21

\begin{align} a_1 c_1 + a_2 c_2 + \cdots + a_n c_n &= a_1 (c_1 - c_2)+ (a_1+a_2)(c_2-c_3)+(a_1+a_2+a_3)(c_3-c_4)+\cdots \\ & \cdots + (a_1+a_2+\cdots+a_{n-1})(c_{n-1}-c_n)+(a_1+a_2+\cdots+a_{n})c_n \end{align}
Then use $c_k - c_{k+1} \ge 0\,$, $c_n \ge 0\,$ and $\sum_{i=1}^k a_i \ge \sum _{i=1}^k b_i\,$.