### THE BOHR-TYPE INEQUALITIES FOR HOLOMORPHIC MAPPINGS WITH A LACUNARY SERIES IN SEVERAL COMPLEX VARIABLES*

Rouyuan Lin1, Mingsheng Liu1,†, Saminathan Ponnusamy2

1. 1. School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China;
2. Department of Mathematics,Indian Institute of Technology Madras, Chennai-600036, India Department of Mathematics, Petrozavodsk State University, ul., Lenina 33, 185910 Petrozavodsk, Russia
• Received:2021-09-16 Revised:2022-06-21 Published:2023-03-01
• Contact: †Mingsheng LIU.E-mail: liumsh@scnu.edu.cn
• About author:Rouyuan Lin, E-mail: 740669790@qq.com; Saminathan Ponnusamy, E-mail: samy@iitm.ac.in
• Supported by:
*Guangdong Natural Science Foundations (2021A1515010058).

Abstract: In this paper, we mainly use the Fréchet derivative to extend the Bohr inequality with a lacunary series to the higher-dimensional space, namely, mappings from $U^n$ to $U$ (resp. $U^n$ to $U^n$). In addition, we discuss whether or not there is a constant term in $f$, and we obtain two redefined Bohr inequalities in $U^n$. Finally, we redefine the Bohr inequality of the odd and even terms of the analytic function $f$ so as to obtain conclusions for two different higher-dimensional alternating series.

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