# $f$ is an entire function, and $1 \leq |f(z)-i|$ for every $z \in \mathbb C$. Show that $f$ is constant.

$f$ is an entire function, and $1 \leq |f(z)-i|$ for every $z \in \mathbb C$. Show that $f$ is constant.

I don't know how to approach this. I tried writing $f(z)= \Sigma _{n=0}^\infty a_nz^n$, but didn't really see how can I proceed.

• Liouville's theorem states that any bounded entire function must be constant. Use it for $1/(f(z)-i)$, which is also entire function – Michael Galuza Jun 6 '15 at 12:35
Observe, that $$g(z)=\frac{1}{f(z)-i}$$ is also entire analytic, and $\lvert g(z)\rvert\le 1$.
Hence $g$ is constant, and so is $f$.