# Zeros of holomorphic functions in the unit disk and \(\rho \)-trigonometrically convex functions

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## Abstract

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

## Keywords

Holomorphic function Zero set Subharmonic function Riesz measure Uniqueness theorem \(\rho \)-Trigonometrically convex function## Mathematics Subject Classification

Primary 30C15 31A05 Secondary 31A15## Notes

### Acknowledgements

The authors thank the organizers of International Conferences “Complex Analysis and Related Topics 2018” (April 23–27, 2018, Euler International Mathematical Institute, St. Petersburg, Russia) and “XXVII St.Petersburg Summer Meeting in Mathematical Analysis” (August 6–11, 2018, St. Petersburg, Russia) for the invitation and for the opportunity to report the results related to the content of this article. The authors are also very grateful to the reviewer for useful comments and corrections.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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