Asked Mar 7, 2019

Suppose that f(z) is an entire function such that f(z)/zn is bounded for |z|>= R. Show that f(z) is a polynomial of degree at most n. 


Expert Answer

Step 1

Let f(z) is an entire function such that f(z)/zn is bounded for |z|≥ R.

Since is an entire function, it has a power series about the origin which converges for all z.

Step 2

We will show that f(k)(0) = 0 for k>n. This will implies that ak=0 for k>n. And thus the power series is just a polynomial.

Consider a circle of radius R, then Cauchy\'s inequality says that

Step 3

Then by the hyp...


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