Integrals of differences of subharmonic functions. I. An integral inequality with Nevanlinna characteristic and modulus of continuity of measure

2021/05/25 by Б. Н. Хабибуллин, Khabibullin, B. N.
Mathematics · Medicine · #30D35 (Primary) 31A05 #31B05 (Secondary) #Complex Variables (math.CV) #Endometriosis Research and Treatment #FOS: Mathematics #Holomorphic and Operator Theory #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2106.08947

openalex publication_date 2021/05/25 · openalex created_date 2021/06/22 · openalex updated_date 2026/07/28

Abstract

We obtain new integral inequalities for the integrals of the difference of subharmonic functions in measure through their Nevanlinna characteristic and some functional characteristic of the measure. These results are new also for meromorphic functions. Let us illustrate the main result for the case of a meromorphic function. We denote by D_z(r) a closed disc of radius r centered at z in the complex plane \mathbb C. Let μ≥ 0 be a Borel measure on D₀(r), and M:=μ(D₀(r))r with f(0)∈ \mathbb C and the Nevanlinna characteristic T_f . Then there is ∫ ln⁺|f| dμ≤ 2(R+r)/(R-r)T_f(R)M\biggl(1+∫₀^R+r(h(t))/(t) dt\biggr).

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