## Complex #26: polynomial properties

**Problem #26:** If is an entire function that is not a polynomial, prove that, given arbitrary , , and integer , there exists a such that and .

The way this is worded makes my head hurt, so let’s formulate it equivalently as Not B implies Not A instead of A implies B.

**Reformulation:** Let be an entire function. Suppose that there exists a , , and an integer such that whenever . Prove that is a polynomial.

**Proof:** By the given and a limiting process, for . By the maximum modulus principle, the same fact is true for . By the Cauchy-Lagrange inequalities if , we have , where is the maximum value of for . Since this circle is completely contained in the disk , , and so we have for and thus for . By Liouville’s theorem, is a constant, and thus is a polynomial.