Meromorphic functions and differences of subharmonic functions in integrals and the difference characteristic of Nevanlinna. II. Explicit estimates of the integral of the radial maximum growth characteristic

2021/04/16 by Б. Н. Хабибуллин, Khabibullin, B. N.
Mathematics · Medicine · #Meromorphic and Entire Functions #Holomorphic and Operator Theory #Endometriosis Research and Treatment

paper · pdf · doi:10.48550/arxiv.2104.11086

Abstract

Let U\not≡ ±∞ be the difference of subharmonic functions, i.e., a δ-subharmonic function, on a closed disc of radius R centered at zero. In the preceding first part of our paper, we obtained general estimates for the integral of the positive part of the radial maximum growth characteristic \mathsf M_U(t):=sup\U(z)\bigm| |z|=r\ over the increasing integration function m on the segment [0, r] via the Nevanlinna difference characteristic and the modulus of continuity of the function m. The second part of the work gives an explicit view for such estimates, provided that the modulus of continuity of the function m does not exceed some differentiable function h on the open interval (0,r) with the only condition sup_{t∈ (0,r)}\dfrach(t)th'(t)0. The estimates are optimal in a certain sense.

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