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for questions involving inequalities, upper and lower bounds.
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Upper bound of $\frac{\sum_i c_ia_ie_i}{\sum_i d_ib_if_i}$?
Let $\sum_i c_i =\sum_i d_i=1$, where $c_i,d_i \ge 0$. Assume that $\frac{\sum_i c_ia_i}{\sum_i d_ib_i} \le \epsilon_1$ and $\frac{\sum_i c_ie_i}{\sum_i d_if_i} \le \epsilon_2$, where $a_i,b_i,e_i,f_i …