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I have two real sequences $a_1,a_2,\dots,a_n$ and $b_1, b_2, \dots, b_n$, with $a_i > 0$ and $1 \leq b_i < n$, and I'm looking for a lower bound of $\sum_i \frac{a_i}{b_i}$ in terms of $\sum_i a_i$ and $\sum_i b_i$. There is also an extra contraint that if $b_i$ is large then $a_i$ is small (something like $a_i < b_i/e^{b_i})$

They're not necessarily monotonic, so I can't use Chebyshev's inequality. Is there something else that I'm missing?

Edit:

After thinking a bit more about the problem I get that, as stated, there are instances where $\sum \frac{a_i}{b_i} = \frac{\sum a_i}{\sum b_i - (n-1)}$.

That is if $a_i = \epsilon$ for $1 \leq i \leq n-1$ and $a_n = \sum a_i - (n-1)\epsilon$, and $b_i = 1$ for $1 \leq i \leq n-1$ and $b_n = \sum b_i - (n-1)$. Then, as long as $\sum a_i < \frac{\sum b_i - (n-1)}{e^{\sum b_i - (n-1)}} + (n-1)\epsilon$, we get the above as $\epsilon$ goes to $0$. Is this tight as long as the inequality is satisfied? If it isn't satisfied, then $\sum a_i$ is larger than $\frac{\sum b_i - (n-1)}{e^{\sum b_i - (n-1)}}$, implying that the lower bound should be larger.

So to tweak the problem a bit more, what if the inequality isn't satisfied (for small $\epsilon$)? Say for $\sum a_i = 1$ and $\sum b_i = 2n$?

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# Ratio of Sequences Sum Inequality

I have two real sequences $a_1,a_2,\dots,a_n$ and $b_1, b_2, \dots, b_n$, with $a_i > 0$ and $1 \leq b_i < n$, and I'm looking for a lower bound of $\sum_i \frac{a_i}{b_i}$ in terms of $\sum_i a_i$ and $\sum_i b_i$. There is also an extra contraint that if $b_i$ is large then $a_i$ is small (something like $a_i < b_i/e^{b_i})$

They're not necessarily monotonic, so I can't use Chebyshev's inequality. Is there something else that I'm missing?