I am trying to write the inverse Mellin transform of $\sin(x)f(x)$ in terms of the inverse Mellin transform of $f(x)$.
That is, suppose that $F(y)$ is the inverse Mellin transform of $f(x)$, $$f(x) = \int_{0}^{\infty} y^{x-1}F(y) dy.$$
How can I write the inverse Mellin transform of $\sin(x)f(x)$ in terms of $F(y)$?