2020/12/14 by Б. Н. Хабибуллин, Khabibullin, B. N., F. B. Khabibullin +1
Mathematics · #30C15 (Primary) #30D20 (Secondary) #31A05 #Complex Variables (math.CV) #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2012.12660
openalex publication_date 2020/12/14 · openalex created_date 2021/01/05 · openalex updated_date 2026/07/28
Let D be a domain in the complex plane, M be an extended real function on D. If f is a non-zero holomorphic function on D with an upper constraint |f|≤ exp M on this domain D, then it is natural to expect that there must be some upper constraints on the distribution of zeros of this holomorphic function exclusively in terms of the function M and the geometry of the domain D. We have investigated this question in detail in our previous works in the case when M is a subharmonic function and the domain D is arbitrary or with a non-polar boundary. The answer was given in terms of limiting the distribution of zeros of f from above via the Riesz measure of the subharmonic function M. In this article, the function M is the difference of subharmonic functions, or a δ-subharmonic function, and the upper constraints are given in terms of the Riesz charge of this δ-subharmonic function M. These results are also new to a certain extent for the subharmonic function M. The case when the domain D is the complex plane is considered separately. For the complex plane, it is possible to reach the criterion level.