Is it possible to build any condition on $x,y,z \in \Bbb R$ for $x\log(x)+y\log(y)<2z\log(z)$?

Question

I am trying to come up with a condition on $x,y,z \in \Bbb R$; where $x,y,z\in [0,1]$ for the following equation (the base of log is 2).

$$x\log(x)+y\log(y)<2z\log(z)$$

Can I come up with a relation between $(x,y)$ and $z$? Any hints will help.

Note that: We are considering: $0\log 0=1\log 1=0$.

Answer 1

Since$$x\log x+y\log y=\log(x^x)+\log(y^y)=\log(x^xy^y)$$and$$2z\log z=\log(z^{2z}),$$your condition is equivalent to $x^xy^y<z^{2z}$.