I have two functions whose functional forms are of Shannon entropy type. The functions are:
$$f_1 = x \log x + (1-x)\log (1-x)+y \log y + (1-y)\log (1-y)$$ and $$f_2 = xy \log (xy) + (1-xy)\log (1-xy)$$ Here both $x,y$ are less than 1. I want to know whether $f_2>f_1$ or $f_1>f_2$ ? Is there any inequality which says that. I tried some analytical calculation but couldn't succeed. Can anyone suggest anything ?