Suppose n belongsto N and r c belongsto R0 Let f be an entri

Suppose n belongsto N, and r, c belongsto R_>0. Let f be an entrire function such that |f(z)| lessthanorequalto c|z^n| for all z belongsto C with |z| greaterthanorequalto r. Then f is a polynomial of degree at most n.

Solution

We may assume f(0) not equal to zero (by considering g(z)=f(z) +1, say.).

Consider the function

h(z)=f(z)/zⁿ . as f is entire, so is h(z).

Clearly h(z) is bounded inside |z|< r and is given to be bounded outside |z|<r.

By Liouville, h(z) is constant. Hence the result