Help me please.

Problem:

1)Let $\displaystyle f$ be an analityc function on the unit disk :$\displaystyle D=\left \{ \left | z \right |<1 \right \}$ and $\displaystyle f^2$ - polynom in domain $\displaystyle D$.

Is $\displaystyle f$ analityc function in $\displaystyle D$?

2)Let $\displaystyle f$ be an entire function and $\displaystyle f^2$ polynom.

Is$\displaystyle f$ polynom ?

Thanks a lot.