SCHWEITZER 1977 Problem 6

Let $f$ be a real function defined on the positive half-axis for which $f(xy)=xf(y)+yf(x)$ and $f(x+1) \leq f(x)$ hold for every positive $x$ and $y$ . Show that if $f(1/2)=1/2$ , then $$f(x)+f(1-x) \geq -x \log_2 x -(1-x) \log_2 (1-x)$$ for every $x\in (0,1)$ .

Z. Daroczy, Gy. Maksa