The restriction from below of the subharmonic function by the logarithm of the module of entire function

2022/03/22 by Б. Н. Хабибуллин, Khabibullin, B. N.
Mathematics · #28A78 ( Secondary) #30D15 #30D20 (Primary) 31A05 #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2203.12383

openalex publication_date 2022/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let u\not≡ -∞ be a subharmonic function on the complex plane \mathbb C. Then for any function r\colon\mathbb C→ (0,1] satisfying the condition inf_{z∈\mathbb C}(ln r(z))/(ln(2+|z|))gt;-∞, there is an entire function f\not≡ 0 such that ln |f(z)|≤ (1)/(2π)∫₀^2πu(z+r(z)e^iθ) \mathrm dθ\quadfor all z∈\mathbb C. A similar result is established for subharmonic functions of finite order with inequalities of the form ln|f(z)|≤ u(z) at all points z∈\mathbb C∖ E, where the exceptional set E is small in terms of d-dimensional Hausdorff content of E with variable radius r.

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