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! :)