• Complex Analysis: #31 Functions of Finite Order with Zeros

Theorem 53
Let f be an entire function of strict order α. Then there exists a C > 0 such that, for all R > 0 and D_R = {z ∈ ℂ : |z| ≤ r}, we have

v_f(R) ≤ CR^α, 

where v_f(R) is the number of zeros of f in D_R.

Proof
Assume first that f(0) ≠ 0. According to Jensen’s inequality (leaving out the part where we integrate from 0, and remembering that |f(z)| ≤ Ke^|z|α , for some K > 0), we have, for R > 1

Finally, if we do have f(0) = 0, then take g(z) = f(z)/z^m, where m is the order of the zero at 0. Then the theorem will apply to g, and since m is then fixed, to f as well.


Theorem 54
Again, f is an entire function of strict order α with zeros {z_n}, listed (with multiplicity) in order of increasing absolute value. We assume that f(0) ≠ 0. Then for every δ > 0 we have the series

∑ |z_n|^−α−δ

converging.

Proof
Using partial summation and the previous theorem, where we assume that R ∈ ℕ, we have

and this last sum is convergent.

This, combined with the discussion concerning the genus of a canonical product of discrete elements of ℂ (see section 24), leads to:

Corollary (Hadamard)
Let f be an entire function of finite order α with zeros {z_n}_n∈ℕ. Then

IMPORTANT NOTE:
This series on Complex Analysis has been taken from the lecture notes prepared by Geoffrey Hemion.  The document can be found at his homepage.