my Complex Analysis final is in a couple of days and i'm struggling with this question -
"Is there an entire function $f$ that satisfies $|f(z)| = |z| + 1$ for every $z$ in the complex plane for which $|z| \ge 2017$?"
I deducted if such function exists, it must be a polynomial. I tried tinkering around with $1/f$ and $f(1/z)$ but didn't really find anything useful, also tried using Rouché's theorem but I can't seem to prove/disprove the existence of a function.
I'd love a hint! :)
You already know that $f$ is a polynomial function. Let $n$ be its degree. Then$$\lim_{\lvert z\rvert\to\infty}\frac{\lvert f(z)\rvert}{\lvert z\rvert^n}=1.$$But then it follows from your hypothesis that $n=1$ and that $f(z)=az+b$ with $\lvert a\rvert=1$. However there are no such numbers $a$ and $b$ so that $\lvert az+b\rvert=\lvert z\rvert+1$ when $\lvert z\rvert$ is large enough.