$f(z)$ is an entire function , if $|f(z)|\le 100\log |z|$ for each $z$ with $|z| \ge 2$, find the value of $f(1)$

Question

Let $f$ be an entire function on $\mathbb{C}$ such that $|f(z)|\le 100\log|z|$ for each $z$ with $|z|\ge 2$.If $f(i)=2i$ then what is the value of $f(1)$ ?
How to solve this kind of question? I know that entire functions are those which don't have singularities on the finite plane, but how can this fact be helpful in solving this question?

Answer 1

There are several reasonable approaches. Here are a couple:

1. From the bound, $\dfrac{f(z)}{z}$ has a removable singularity at $z = \infty$, so $f$ must be a polynomial of degree at most $1$. But then the bound shows that the degree of $f$ must in fact be $0$, so $f$ is constant.

2. Use Cauchy's integral theorem for $f'$ on a large circle of radius $R$ centered at $z_0$. Let $R \to \infty$ and make standard estimates on the integral to show that $f'(z_0) = 0$. Hence $f$ is constant.