Imagine I have three sets of strictly positive real numbers: $a_i,b_i,c_i>0$, $\forall i=1,\ldots,n$. For finite $n$. And further that the following inequality holds: \begin{align} \sum_i a_i \leq \sum_i b_i + \sum_i c_i \end{align} Then does the following inequality hold? \begin{align} \sum_i \log(1+a_i) \leq \sum_i \log(1+b_i) + \sum_i \log(1+c_i) \end{align}
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$\begingroup$ My intuition says 'yes' $\endgroup$– user5954246May 17, 2016 at 21:51
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$\begingroup$ You could represent it : $\prod{(a_{i}+1)} \le \prod{(c_{i}+1)}\prod{(b_{i}+1)} $ $\endgroup$– openspaceMay 17, 2016 at 21:55
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