Bounded complex valued function

[g1990]

Homework Statement

1. f(z) is a function that is analytic on all of the complex plane, and mod(f)<=mod(z). Prove that f=cz.
2. f(z) is analytic on all of the complex plane, and mod(f)<= sqrt(mod(z)). Prove that f is constant

Homework Equations

Liouvilles thm: the only bounded entire functions are constant
Maximum modulus thm: If g is analytic on a domain D and achieves a maximum modulus on the interior of D, then g is constant

The Attempt at a Solution

1. I think I have this one. So, |f(z)/z|<=1, and analytic enerywhere except at z=0. If z=0 is an essential singularity or a pole, f(z)/z is not bounded near z, which is a contradiction. Thus, z=0 is a removable singularity or f(z)/z. So, There is an entire function h(z) where h(z)=f(z)/z when z is not 0, and h(z)=c when z=0 for some value c. Then, |h(z)| is bounded by max(1,c), and is constant. Then, f(z)/z=c for all z not equal to zero, so that f=cz. At zero, f(z)=0 by continuity because f is analytic everywhere on C. thus, f(z)=cz on the complex plane.
2. I don't know where to begin here. I would like to show that f is bounded, and then use Liouville's thm, but I don't know if that's the way to go.

 

[Dick]

Start from |f(z)^2/z|<=1.

 

[Count Iblis]

In case of 2), you can use that the inequality implies that f(0) = 0. Then the fact that f is analytic and thus complex differentiable at zero implies that the limit of f(z)/z for z to zero must exist as it is the derivative of f at z = 0. Then you can complete the proof by splitting the complex plane in some small neighborhood of zero and the rest of the complex plane and considering the behavior of f(z)/z.

You will be led to a stronger conclusion than asked in the problem: f(z) is actually zero.