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Let \mathbb{R} denote the set of all real numbers and \mathbb{R}^+ the subset of all positive ones. Let \alpha and \beta be given elements in \mathbb{R}, not necessarily distinct. Find all functions f: \mathbb{R}^+ \mapsto \mathbb{R} such that

f(x)f(y) = y^{\alpha} f \left( \frac{x}{2} \right) + x^{\beta} f \left( \frac{y}{2} \right) \forall x,y \in \mathbb{R}^+.

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