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2019 | OriginalPaper | Chapter

12. Counting Zeros of Holomorphic Functions

Author : Johannes Sjöstrand

Published in: Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations

Publisher: Springer International Publishing

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Abstract

In this chapter we will generalize Proposition 3.​4.​6 of Hager about counting the zeros of holomorphic functions of exponential growth. In Hager and Sjöstrand (Math Ann 342(1):177–243, 2008. http://​arxiv.​org/​abs/​math/​0601381) we obtained such a generalization, by weakening the regularity assumptions on the functions ϕ. However, due to some logarithmic losses, we were not quite able to recover Hager’s original result, and we still had a fixed domain Γ with smooth boundary.

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Footnotes
1
Here “4” can be replaced by any fixed constant >2.
 
2
In the following we simply write C instead of C new, following the general rule that constants change from time to time.
 
3
That is, given by an equation f(z) = 0, where f belongs to a bounded family of smooth real functions with the property that f(z) = 0 ⇒ \(|\nabla f(z)|\ge 1/{\mathcal {O}}(1)\) uniformly for all f in the family.
 
4
Which says that if \(\Omega \Subset \mathbf {C}\) is a connected open set with smooth boundary and K ⊂ Ω a compact subset, then there exists a constant C = C Ω,K > 0 such that u(z 1) ≤ Cu(z 2) for all z 1, z 2 ∈ K and every non-negative harmonic function u on Ω, uniformly if Ω varies in a family of uniformly bounded subsets of C and \(\mathrm {dist}\,(K,\partial \Omega )\ge 1/{\mathcal {O}}(1)\) uniformly, see e.g., [4, Section 6.3].
 
5
This also follows more directly from the spherical mean value inequality for subharmonic functions.
 
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Metadata
Title
Counting Zeros of Holomorphic Functions
Author
Johannes Sjöstrand
Copyright Year
2019
Book
Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations

Print ISBN: 978-3-030-10818-2

Electronic ISBN: 978-3-030-10819-9

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-10819-9_12

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