I'm stuck on a qualifying exam question, not really familiar with the method of solution. It goes as follows:
Let $f(z)$ be an an entire function and $g(z)$ be analytic in a neighborhood of $z=1$ which satisfies $g^{(n)}(1)=(f^{(n)}(1))^{\alpha}/(n!)^{\alpha-1}$, where $\alpha >0$. Show that $g(z)$ can be extended to an entire function.
Any help would be greatly appreciated.
edit: fixed typo