# On the number of zeros of functions in analytic quasianalytic classes

## TL;DR

This paper establishes an explicit estimate on the number of zeros of functions in general analytic quasianalytic classes, extending previous results from Gevrey classes through a reduction to classical quasianalyticity.

## Contribution

It provides a new zero-count estimate for analytic quasianalytic classes, generalizing prior Gevrey class results via a reduction to classical quasianalyticity.

## Key findings

- Derived an explicit zero estimate for general quasianalytic classes.
- Extended previous Gevrey class results to broader analytic classes.
- Utilized reduction to classical quasianalyticity problem.

## Abstract

A space of analytic functions in the unit disc with uniformly continuous derivatives is said to be quasianalytic if the boundary value of a non-zero function from the class can not have a zero of infinite multiplicity. Such classes were described in the 1950-s and 1960-s by Carleson, Rodrigues-Salinas and Korenblum. A non-zero function from a quasianalytic space of analytic functions can only have a finite number of zeros in the closed disc. Recently, Borichev, Frank, and Volberg proved an explicit estimate on the number of zeros, for the case of quasianalytic Gevrey classes. Here, an estimate of similar form for general analytic quasianalytic classes is proved using a reduction to the classical quasianalyticity problem.