Find the maximum value of given expression $S = \Sigma f_r.\log(P_r)$

Question

With $\Sigma f_r=1$ and $\Sigma P_r=1$, $\forall f_r, P_r>0$. How to find the maximum value of $S$ where:

$$ S = \Sigma f_r.\log(P_r) $$

Thanks for reading!

Answer 1

I am answering because I don't have enough reputation to comment.

You have $S \leq 0$.

Which means that the maximum of $S$ is $0$. (in the case $f_r = P_r = \delta_{ri}$, when the probability mass functions correspond to Kronecker deltas).

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From your question, I'm wondering if your $S$ shouldn't be the entropy of a probability function. In that case I'll revise my answer.