# 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.

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