2021/04/13 by Б. Н. Хабибуллин, Khabibullin, B. N.
Mathematics · Medicine · #26A15 (Secondary) #30D35 (Primary) 31A05 #Complex Variables (math.CV) #Endometriosis Research and Treatment #FOS: Mathematics #Holomorphic and Operator Theory #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2104.07439
openalex publication_date 2021/04/13 · openalex created_date 2021/04/26 · openalex updated_date 2026/07/28
Let f be a meromorphic function on the complex plane \mathbb C with the maximum function of its modulus M(r,f) on circles centered at zero of radius r. A number of classical, well-known and widely used results allow us to estimate from above the integrals of the positive part of the logarithm ln+M(t,f) over subsets of E⊂ [0,r] via the Nevanlinna characteristic T(r,f) and the linear Lebesgue measure of the set E. The paper gives similar estimates for the Lebesgue-Stieltjes integrals of ln+M (t,f) over the increasing integration function of m. The main part of the presentation is conducted immediately for the differences of subharmonic functions on closed discs with the center at zero, i.e, δ-subharmonic functions. The only condition in the main theorem is the Dini condition for the modulus of continuity of the integration function m. This condition is, in a sense, necessary. Thus, the first part of the work to a certain extent completes in a general form the research on the upper estimates of the integrals of the radial maximum growth characteristics of arbitrary meromorphic and δ-subharmonic functions through the Nevanlinna characteristic with its versions and through the characteristics of the integration function m.