# Zeros of holomorphic functions in the unit disk and $$\rho$$ ρ -trigonometrically convex functions

Zeros of holomorphic functions in the unit disk and $$\rho$$ ρ -trigonometrically convex... Let M be a subharmonic function with Riesz measure $$\mu _M$$ μ M on the unit disk $${\mathbb {D}}$$ D in the complex plane $${\mathbb {C}}$$ C . Let f be a nonzero holomorphic function on $${\mathbb {D}}$$ D such that f vanishes on $${\textsf {Z}}\subset {\mathbb {D}}$$ Z ⊂ D , and satisfies $$|f| \le \exp M$$ | f | ≤ exp M on $${\mathbb {D}}$$ D . Then restrictions on the growth of $$\mu _M$$ μ M near the boundary of D imply certain restrictions on the distribution of $$\mathsf Z$$ Z . We give a quantitative study of this phenomenon in terms of special non-radial test functions constructed using $$\rho$$ ρ -trigonometrically convex functions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

# Zeros of holomorphic functions in the unit disk and $$\rho$$ ρ -trigonometrically convex functions

, Volume 9 (3) – Feb 2, 2019
12 pages

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Publisher
Springer Journals
Copyright
Copyright © 2019 by Springer Nature Switzerland AG
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-019-00282-1
Publisher site
See Article on Publisher Site

### Abstract

Let M be a subharmonic function with Riesz measure $$\mu _M$$ μ M on the unit disk $${\mathbb {D}}$$ D in the complex plane $${\mathbb {C}}$$ C . Let f be a nonzero holomorphic function on $${\mathbb {D}}$$ D such that f vanishes on $${\textsf {Z}}\subset {\mathbb {D}}$$ Z ⊂ D , and satisfies $$|f| \le \exp M$$ | f | ≤ exp M on $${\mathbb {D}}$$ D . Then restrictions on the growth of $$\mu _M$$ μ M near the boundary of D imply certain restrictions on the distribution of $$\mathsf Z$$ Z . We give a quantitative study of this phenomenon in terms of special non-radial test functions constructed using $$\rho$$ ρ -trigonometrically convex functions.

### Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Feb 2, 2019