Let $p_i \in [0,1]$ and $\sum_{i} p_i = 1$. Let, and furthermore let $a_i$ and $b_i$ be positive real numbers. Is the inequality of $\sum_{i} p_i \frac{a_i}{b_i} \leq \frac{\sum_{i} p_i a_i}{\sum_{i} p_i b_i}$ held $$ \sum_{i} p_i \frac{a_i}{b_i} \leq \frac{\sum_{i} p_i a_i}{\sum_{i} p_i b_i}$$ true?
R. van Dobben de Bruyn
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