I have two vectors $a$ and $b$. I have the two following quantities, $\sum_i a_i \frac{1}{b_i}$ and $\sum_i a_i \frac{1}{\sum_j b_j}$. I know that for every $i$, $0\leq a_i \leq b_i \leq 1$. Which inequality holds between the two sums?
I know that, calling $c_i = 1 / b_i$, the inverse holder inequality holds, that means, $\sum_i a_i \frac{1}{b_i} =|| a c||_1 \leq ||a||_2 ||c||_{-1}$, and I also know that $|| a||_2 \leq || a||_1$, by inclusion of the Lp spaces. Is it possible to show that $\sum_i a_i \frac{1}{b_i} \leq \sum_i a_i \frac{1}{\sum b_i}$ or that $\sum_i a_i \frac{1}{b_i} \geq \sum_i a_i \frac{1}{\sum b_i} ?$ (I am not sure wether one of those is true...)