Abstract:
Let MM be a subharmonic function with Riesz measure νMνM in a domain DD in the nn-dimensional complex Euclidean space Cn, and let f be a nonzero function that is holomorphic in D, vanishes on a set Z⊂D, and satisfies |f|⩽expM on D. Then restrictions on the growth of νM near the boundary of D imply certain restrictions on the dimensions or the area/volume of Z. We give a quantitative study of this phenomenon in the subharmonic framework.
Citation:
B. N. Khabibullin, A. P. Rozit, “On the Distribution of Zero Sets of Holomorphic Functions”, Funktsional. Anal. i Prilozhen., 52:1 (2018), 26–42; Funct. Anal. Appl., 52:1 (2018), 21–34
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\by B.~N.~Khabibullin, A.~P.~Rozit
\paper On the Distribution of Zero Sets of Holomorphic Functions
\jour Funktsional. Anal. i Prilozhen.
\yr 2018
\vol 52
\issue 1
\pages 26--42
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\jour Funct. Anal. Appl.
\yr 2018
\vol 52
\issue 1
\pages 21--34
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https://doi.org/10.4213/faa3485
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This publication is cited in the following 22 articles:
B. N. Khabibullin, “Distributions of zeros and masses of entire and
subharmonic functions with restrictions on their growth along the strip”, Izv. Math., 88:1 (2024), 133–193
B. N. Khabibullin, E. B. Menshikova, “Integral Formulas for Subharmonic and Meromorphic Functions and Completeness of Exponential Systems”, Lobachevskii J Math, 45:1 (2024), 434
B. N. Khabibullin, “Representations on an open set of potentials that are harmonic and coincident outside a compact subset”, Lobachevskii J. Math., 44:4 (2023), 1350
B. Khabibullin, S. V. Kislyakov, “Nevanlinna characteristic and integral inequalities with maximal radial characteristic for meromorphic functions and for differences of subharmonic functions”, St. Petersburg Math. J., 34:2 (2023), 247
B. N. Khabibullin, “Integrals of a difference of subharmonic functions against measures and the Nevanlinna characteristic”, Sb. Math., 213:5 (2022), 694–733
E. B. Menshikova, “Integral formulas of the type of Carleman and B. Ya. Levin for meromorphic and subharmonic functions”, Russian Math. (Iz. VUZ), 66:6 (2022), 28–42
B. N. Khabibullin, E. B. Menshikova, “Preorders on subharmonic functions and measures with applications to the distribution of zeros of holomorphic functions”, Lobachevskii J. Math., 43:3 (2022), 587–611
B. N. Khabibullin, “Poisson–Jensen formulas and balayage of measures”, Eurasian Math. J., 12:4 (2021), 53–73
B. N. Khabibullin, E. B. Menshikova, “Balayage of measures with respect to polynomials and logarithmic kernels on the complex plane”, Lobachevskii J. Math., 42:12 (2021), 2823–2833
B. N. Khabibullin, F. B. Khabibullin, “Necessary and sufficient conditions for zero subsets of holomorphic functions with upper constraints in planar domains”, Lobachevskii J. Math., 42:4, SI (2021), 800–810
B. N. Khabibullin, “Integrals with a meromorphic function or the difference of subharmonic functions over discs and planar small sets”, Lobachevskii J. Math., 42:6, SI (2021), 1175–1182
B. N. Khabibullin, A. V. Shmeleva, Z. F. Abdullina, “Balayage of measures and subharmonic functions to a system of rays. II. Balayages of finite genus and growth regularity on a single ray”, St. Petersburg Math. J., 32:1 (2021), 155–181
E. B. Menshikova, B. N. Khabibullin, “A criterion for the sequence of roots of holomorphic function with restrictions on its growth”, Russian Math. (Iz. VUZ), 64:5 (2020), 49–55
Chyzhykov I.E., “Asymptotic Behaviour of Pth Means of Analytic and Subharmonic Functions in the Unit Disc and Angular Distribution of Zeros”, Isr. J. Math., 236:2 (2020), 931–957
B.N. Khabibullin, “Integrals of subharmonic functions and their differences with weight over small sets on a ray”, Mat. Stud., 54:2 (2020), 162
E. B. Menshikova, B. N. Khabibullin, “K raspredeleniyu nulevykh mnozhestv golomorfnykh funktsii. II”, Funkts. analiz i ego pril., 53:1 (2019), 84–87
B. N. Khabibullin, F. B. Khabibullin, “K raspredeleniyu nulevykh mnozhestv golomorfnykh funktsii. III. Teoremy obrascheniya”, Funkts. analiz i ego pril., 53:2 (2019), 42–58
B. N. Khabibullin, F. B. Khabibullin, “Zeros of holomorphic functions in the unit ball and subspherical functions”, Lobachevskii J. Math., 40:5, 1, SI (2019), 648–659
B. N. Khabibullin, F. B. Khabibullin, “Zeros of holomorphic functions in the unit disk and rho-trigonometrically convex functions”, Anal. Math. Phys., 9:3 (2019), 1087–1098
B. N. Khabibullin, F. B. Khabibullin, “On the Distribution of Zero Sets of Holomorphic Functions: III. Converse Theorems”, Funct Anal Its Appl, 53:2 (2019), 110
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