Which of the following functions is/are constant?

Question

Which of the following functions is/are constant ?

1. Let $f(z)$ be an analytic function in extended complex plane.

2. If $g(z)$ is an entire function such that $g(z) = u + i v$ and $u^2 \leq v^2 + 2012$

I know that entire and bounded function is constant. How to apply this theorem to identify correct options.

Answer 1

Both are constant.

In the first case, suppose $f(z)$ is analytic in the extended complex plane. Then, as the extended complex plain is compact, the image of $f$ will be bounded. So the restriction of $f$ to the complex plane is an entire, bounded function, thus it is constant. So $f$ is constant.

In the second case, you can begin by proving that the image of a entire, non constant function $g(z)$ is dense in the plane (suppose it is not, then let $a$ be a point outside the closure of $g(\mathbb{C})$ and apply Liouville´s theorem to $h(z)=\frac{1}{g(z)-a}$).

Using this, it is clear that the image of $g=u+iv$ will contain points with $u^2>v^2+2012$ if it is not constant.