What are the entire functions with bounded modulus on the complex plane?

[g1990]

Homework Statement

Find all entire functions f(z) with the property that |zf(z)|<=1 for all z in C

Homework Equations

The maximum modulus principle says that the only functions that are entire and bounded are constant functions.

The Attempt at a Solution

I know that if f(z) is entire, then zf(z) is also entire. Thus, if it's modulus is bounded on C, then it must be constant. Thus, zf(z)=c, so that f(z)=c/z where c is a constant. But then, f is not entire. Am I doing something wrong? Or is the only function that satisfies this property zero?

 

[Dick]

That looks right to me. Only f(z)=0 works.