MathDB

Let

f

be a real function defined on the positive half-axis for which f(xy)\equal{}xf(y)\plus{}yf(x) and f(x\plus{}1) \leq f(x) hold for every positive

x

and

y

. Show that if f(1/2)\equal{}1/2, then f(x)\plus{}f(1\minus{}x) \geq \minus{}x \log_2 x \minus{}(1\minus{}x) \log_2 (1\minus{}x) for every

x\in (0,1)

. Z. Daroczy, Gy. Maksa

Problem Statement