Is there a tighter upper and lower bound on concavity of $\log(.)$ function?
It is very well known that $\log(\sum_{i}p_ix_i) - \sum_{i}p_i\log(x_i) \geq 0$ whenever $\sum_{j}p_j = 1$. But are there stronger versions of this inequality, of the form $ \log(\sum_{i}p_ix_i) - \sum_{i}p_i\log(x_i) \geq F(p_i)$ ?
It would be desirable that $F(.)$ be independent of $x_i$, but tighter and non-trivial inequalities even without this restriction would be appreciated. Thanks.