How to prove this inequality that $\frac{\sum_ia_ix_i}{\sum_ia_iy_i}\leq\max_i\frac{x_i}{y_i}$?

Question

How to prove $\frac{\sum_{i=1}^na_ix_i}{\sum_{i=1}^na_iy_i}\leq\max_{1\leq i\leq n}\frac{x_i}{y_i}$, where $a_i, x_i,y_i>0$ for all $i$?

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score 1 · Accepted Answer · answered Aug 04 '23 at 14:20

Let j such that $x_j/y_j=max_i (x_i/y_i)$

We have $\frac{\sum_i a_i x_i}{\sum_i a_i y_i}\leq \frac{x_j}{y_j} \Leftrightarrow \sum_i a_ix_iy_j\leq \sum_i a_i y_i x_j$

$\Leftrightarrow \sum_i a_i (y_iy_j)\left(\frac{x_j}{y_j}-\frac{x_i}{y_i}\right) \geq 0$

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