Find all entire functions $f$ such that for all $z\in \mathbb{C}$, $|f(z)|\ge \frac{1}{|z|+1}$
This is one of the past qualifying exams that I was working on and I think that I have to find the function that involved with $f$ that is bounded and use Louiville's theorem to say that the function that is found is constant and conclude something about $f$. I can only think of using $1/f$ so that $\frac{1}{|f(z)|} \le |z|+1$ but $|z|+1$ is not really bounded so I would like to ask you for some hint or idea.
Any hint/ idea would be appreciated.
Thank you in advance.
Best Answer
Hint: As you suggest, you can consider $g = 1/f$, which is itself an entire function (why?). Use the estimate $|g(z)|\leq 1 + |z|$ and the Cauchy estimates to show that $g$ is a (nonvanishing) polynomial of degree $\leq 1$. What can you conclude?