ILL 1992 Problem 48

Find all the functions $f : \mathbb R^+ \to \mathbb R$ satisfying the identity $$f(x)f(y)=y^{\alpha}f\left(\frac x2 \right) + x^{\beta} f\left(\frac y2 \right) \forall x,y \in \mathbb R^+$$ Where $\alpha,\beta$ are given real numbers.