Some new inequalities for the logarithmic map, with applications to entropy and mutual information (Q2755645)

Consider a natural number \(n>1\) and a real number \(c>1\) . Denote by \(\log \) a logarithm taken to base \(c\) and by \(K\) the set \(\{1,2,\dots ,n\}\) . If for each \(i\in K\) the numbers \(a_{i}\) and \(q_{i}\) are positive and \(\sum_{i\in K}q_{i}=1\) , then the inequalities NEWLINE\[NEWLINE 0\leq \sum_{i\in K}q_{i}a_{i}\log a_{i}-\sum_{i\in K}q_{i}a_{i}\log \left( \sum_{i\in K}q_{i}a_{i}\right) \leq \frac{1}{2}\sum_{i,j\in K}q_{i}q_{j}(a_{i}-a_{j})\log \frac{a_{i}}{a_{j}} , NEWLINE\]NEWLINE are valid. This main result of the paper is then used for deriving some new bounds for entropy, conditional entropy and mutual information.