Logarithmic inequality involving minimum

Question

Given two sets $\{ a_1, \dots, a_N \}$, $\{ b_1, \dots, b_N \}$ with $0 < a_i, b_i < 1$ does the following logarithmic inequality always hold? $$ \log \left( \sum_i \left( a_i + b_i - \min(a_i, b_i) \right) \right) \le \log \left( \sum_i a_i \right) + \log \left( \sum_i b_i \right) - \log \left( \sum_i \min(a_i, b_i) \right) $$

Answer 1

Hint: noting that $a_i+b_i-\min(a_i,b_i)=\max(a_i,b_i)$, you can rearrange to find your desired inequality is equivalent to \begin{equation} \sum_{i=1}^n \max(a_i,b_i)\sum_{i=1}^n \min(a_i,b_i)\leq \sum_{i=1}^n a_i\sum_{i=1}^n b_i. \end{equation} Is this always true? As an extra hint: think about minimizing the function $xy$ subject to a constraint that $x+y=c$ for some constant $c>0$. Heuristically, when is this small?