International Journal of Nonlinear Analysis and Applications

Some properties of bicomplex holomorphic functions

Document Type : Research Paper

Authors

1 Department of Mathematics, Semnan University, Semnan, Iran

2 Department of Mathematics, University of Kashmir, Srinagar, 190006, India

Abstract

In this paper, we first establish the bicomplex version of Rouche's theorem. Also, a new approach is given to prove the maximum modulus principle for bicomplex holomorphic functions. Our proof is based on the direct method and extends the result proved by Luna-Elizarraras et al. Finally, we generalize the Hurwitz's theorem to bicomplex space.

Keywords

[1] D. Alpay, M.E. Luna-Elizarraras, M. Shapiro, and D.C. Struppa, Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis, Springer Briefs in Mathematics, 2014.
[2] K.S. Charak and D. Rochon, On the factorization of bicomplex meromorphic functions, Trends in Mathematics, Birkhauser Verlag Basel/Switzerland, 2008, pp. 55–68.
[3] K.S. Charak, D. Rochon, and N. Sharma, Normal families of bicomplex holomorphic functions, Fractals 17 (2009), no. 3.
[4] M.E. Luna-Elizarraras, M. Shapiro, D.C. Struppa, and A. Vajiac, Bicomplex number and their elementary function, CUBO Math. J. 14 (2012), no. 2, 61–80.
[5] M. Marden, Geometry of Polynomials, 2nd ed., Mathematical Surveys, vol. 3, Amer. Math. Soc, Providence, R.I., 1966.
[6] A.A. Pogorui and R.M. Rodriguez-Dagnino, On the set of zeros of bicomplex polynomials, Complex Variab. Elliptic Equ. 51 (2006), no. 7, 725–730.
[7] S. Ponnusamy and H. Silverman, Complex Variables with Applications, Springer Science & Business Media, 2007.
[8] R.D. Poodiack and K.J. LeClair, Fundamental theorems of algebra for the perplexes, College Math. J. 40 (2009), no. 5, 322–335.
[9] G.B. Price, An Introduction to Multicomplex Spaces and Functions, Monographs and Textbooks in Pure and Applied Mathematics, vol. 140, Marcel Dekker, Inc., New York, 1991.
[10] J.D. Riley, Contributions to the theory of functions of a bicomplex variable, Tohoku Math. J. Second Ser. 2 (1953), no. 5, 132–165.
[11] C. Segre, Le rappresentazioni reali delle forme complesse e gli enti iperalgebrici (The real representation of complex elements and hyper algebraic entities), Math. Ann. 40 (1892), 413–467.
International Journal of Nonlinear Analysis and Applications
Volume 16, Issue 5
May 2025
Pages 27-33
  • Receive Date: 01 March 2024
  • Accept Date: 25 May 2024