Journal of Hebei University(Natural Science Edition) >

2023 , Vol. 43 >Issue 2: 127 - 132

DOI: https://doi.org/10.3969/j.issn.1000-1565.2023.02.003

Orders and types of harmonic entire mappings

  • QIAO Jinjing ,
  • TIAN Yue ,
  • ZHENG Lifang
Expand
  • 1. Hebei Key Laboratory of Machine Learning and Computational Intelligence, College of Mathematics and Information Science, Hebei University, Baoding 071002, China; 2. Shijiazhuang Tiedao University Sifang College, Shijiazhuang 051132, China

Received date: 2022-02-13

  Online published: 2023-04-06

Fold

Abstract

In order to discuss the growth estimates of harmonic entire mappings, the order and the type of harmonic entire mappings are studied, and they can be characterized by the coefficients of the analytic parts and anti-analytic parts. For a strictly increasing sequence of positive integers{nk}∞k=1, two quantities are defined by using the values of nk order derivative of the analytic part and anti-analytic part of a harmonic entire mapping at a certain point, and it is proved that these two quantities are equal to the order and the type of the harmonic entire mapping, respectively, almost everywhere.

Key words： entire function; harmonic entire mapping; order; type

Cite this article

QIAO Jinjing , TIAN Yue , ZHENG Lifang . Orders and types of harmonic entire mappings[J]. Journal of Hebei University(Natural Science Edition), 2023 , 43(2) : 127 -132 . DOI: 10.3969/j.issn.1000-1565.2023.02.003

References

[1] BOAS R P. Entire functions [M]. New York: Academic Press, 1954.
[2] DUREN P. Harmonic mappings in the plane [M]. New York: Cambridge Univ Press, 2004.
[3] CHUAQUI M, DUREN P, OSGOOD B. The Schwarzian derivative for harmonic mappings [J]. J Anal Math, 2003, 91(1): 329-351. DOI: 10.1007/BF02788793.
[4] LIU Z, PONNUSAMY S. Bohr radius for subordination and K-quasiconformal harmonic mappings [J]. Bull Malays Math Sci Soc, 2019, 42(5): 2151-2168. DOI: 10.1007/s40840-019-00795-9.
[5] LIU M S, PONNUSAMY S, WANG J. Bohr’s phenomenon for the classes of Quasi-subordination and K-quasiregular harmonic mappings[J]. RACSAM, 2020, 114(3): 115. DOI: 10.1007/s13398-020-00844-0.
[6] LEWY H. On the non-vanishing of the Jacobian in certain one-to-one mappings [J]. Bull Amer Math Soc, 1936, 42(10): 689-692. DOI: 10.1090/s0002-9904-1936-06397-4.
[7] DENG H, PONNUSAMY S, QIAO J, et al. On harmonic entire mappings [J]. RACSAM, 2021, 55: 619-638. DOI:10.1007/s13398-021-01148-7.
[8] PAPANICOLAOU V G, KALLITSI E, SMYRLIS G. On the order and the type of an entire function [EB/OL].(2021-09-06)[2022-02-13]. https://arxiv.org/abs/2109.02535. (
Options
Outlines

/

〈 〉