integral equation solution for two functions $ f(x) $ and $ g(x) $ and see if they are related

Question

given two functios $ f(x) $ and $ g(x) $ related by $$\frac{ \Gamma(s-1/2)}{\Gamma(s) \sqrt{ \pi}}\int_{0}^{\infty}dx \frac{g(x)dx}{(x+y)^{s-1/2}}=\int_{0}^{\infty}dx \frac{f(x)dx}{(x+y)^{s}}$$ what relation exists between them ? I believe that
$$ g(x)= A \frac{d^{1/2}f(x)}{dx^{1/2}}$$ for some constant $A$ but I am not sure.