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Since $f$ is meromorphic on $\mathbb{C}$, $f$ is holomorphic on $\mathbb{C}$ except for some isolated singularities which are poles. Aslo, $f$ being meromorphic we can write $f$ as $g(z)/h(z)$ both holomorphic with $h\neq 0$.

Now does multiplying through help lead to the conclusion?

So we would have $fh = g$. If so, I am not sure with what to do next.