I am deriving some bounds involving two functions, which are $f(x)=\log(x)$ and $f(x)=-x\times\log(x)$. I found in several books that these two functions are concave. I have draw them down using python to quickly check. For that reason, by applying Jensen's inequality the next inequality should hold:
$f(x)+f(y)\leq f(x+y)$
which holds in case of $f(x)=\log(x)$ but not for $f(x)=-x\times\log(x)$. What am I missing?
Thank you.
Here is a possible way I think, comment if anything wrong.
Note that, for any $x>1$ we have $(x+y)^{x+y}=(x+y)^x(x+y)^y>x^xy^y$
So, $$(x+y)^{-(x+y)}<x^{-x}y^{-y}$$ $$\implies -(x+y)\log(x+y)<-x\log x-y\log y$$ $$\implies f(x+y)<f(x)+f(y)$$