Suppose $f$ is an entire function satisfying any one of the following conditions for all $z\in \mathbb C$
(1) im$f(z)$ has no zeros
Then f is constant.
My thought: For(2) since $|f(z)|\geq 1$ ,then $f$ has no zero in $\mathbb C$.
Define $g=1/f$, then $g$ is bounded entire function implies $g$ is constant implies $f$ is constant, am I right ? I have no idea about (1), please give some hints. Thanks.