I had these two problems on a problem set and my solutions were incorrect and I haven't been able to figure them out.

Let f(z) be analytic on the complex plane. Prove f is a constant in the following cases:

$\displaystyle |f(z)|\geq7$

and

$\displaystyle Re f(z)\geq 7$, hint: consider $\displaystyle \exp(f(z))$

both for all z in the complex plane.

Thanks.