# Integrals of subharmonic functions and their differences with weight over small sets on a ray

### Abstract

Let $E$ be a measurable subset in a segment $[0,r]$ in the positive part of the real axis in the complex plane, and $U=u-v$ be the difference of subharmonic functions $u\not\equiv -\infty$ and $v\not\equiv -\infty$ on the complex plane. An integral of the maximum on circles centered at zero of $U^+:=\sup\{0,U\} $ or $|u|$ over $E$ with a function-multiplier $g\in L^p(E) $ in the integrand is estimated, respectively, in terms of the characteristic function $T_U$ of $U$ or the maximum of $u$ on circles centered at zero, and also in terms of the linear Lebesgue measure of $E$ and the $ L^p$-norm of $g$. Our main theorem develops the proof of one of the classical theorems of Rolf Nevanlinna in the case $E=[0,R]$, given in the classical monograph by Anatoly A. Goldberg and Iossif V. Ostrovsky, and also generalizes analogs of the Edrei-Fuchs Lemma on small arcs for small intervals from the works of A. F. Grishin, M. L. Sodin, T. I. Malyutina. Our estimates are uniform in the sense that the constants in these estimates do not depend on $U$ or $u$, provided that $U$ has an integral normalization near zero or $u(0)\geq 0$, respectively.

### References

R. Nevanlinna, Le théoremè de Picard–Borel et la théorie des fonctions méromorphes, Paris: Gauthier-Villars, 1929.

A.A. Goldberg, I.V. Ostrovskii, Value distribution of meromorphic functions. With an appendix by Alexandre Eremenko and James K. Langley, Translations of Mathematical Monographs, V.236, AMS, Providence, RI, 2008.

A. Edrei, W.H.J. Fuchs, Bounds for number of deficient values of certain classes of meromorphic functions, Proc. London Math. Soc., 12 (1962), 315–344.

A.F. Grishin, M.L. Sodin, Growth along a ray, distribution of roots with respect to arguments of an entire function of finite order, and a uniqueness theorem, Republican collection “Function theory, functional analysis and their applications”, 50 (1988), 47–61.

A.F. Grishin, T.I. Malyutina, New formulas for inidicators of subharmonic functions, Mat. Fiz. Anal. Geom., 12 (2005), No1, 25–72.

B.N. Khabibullin, A.V. Shmeleva, Z.F. Abdullina, Balayage of measures and subharmonic functions on a system of rays. II. Balayage of finite genus and regularity of growth on one ray, Algebra i Analiz, 32 (2020), No1, 208–243.

L.A. Gabdrakhmanova, B.N. Khabibullin, A theorem on small intervals for subharmonic functions, Russ. Math., 64 (2020), 12–20. https://doi.org/10.3103/S1066369X20090029

Th. Ransford, Potential theory in the complex plane, Cambridge: Cambridge University Press, 1995.

W.K. Hayman, P.B. Kennedy, Subharmonic functions, V.I, London Math. Soc. Monogr., 9, London–New York: Academic Press, 1976.

M.G. Arsove, Functions representable as differences of subharmonic functions, Trans. Amer. Math. Soc., 75 (1953), 327–365.

M.G. Arsove, Functions of potential type, Trans. Amer. Math. Soc., 75 (1953), 526–551.

A.F. Grishin, Nguyen Van Quynh, I.V. Poedintseva, Representation theorems of δ-subharmonic functions, Visnyk of V.N. Karazin Kharkiv National University. Ser. “Mathematics, applied mathematics and mechanics”, 1133 (2014), 56–75.

B.N. Khabibullin, A.P. Rozit, On the distribution of zero sets of holomorphic functions, Funct. Anal. Appl., 52 (2018), No1, 21–34.

Copyright (c) 2020 Bulat Khabibullin

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Matematychni Studii is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) license.