Upperbound for $ \sum _{i=1}^N a_i\ln a_i $

Question

It's easy to prove that following upperbound is true:

$\sum_{i=1}^N a_i\ln a_i \le A \ln A$, where $\sum_{i=1}^N a_i=A$ and $ a_i\ge 1$

I'm wondering, is there stronger upperbound?

Answer 1

You can also use the following estimate $$\sum_{j=1}^{N} a_j\ln a_j \leq A\ln (A-N+1) .$$