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## 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?