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\begin{document}

\title{The Schwarz lemma in bicomplex analysis}



\author[1]{Zekun Li}%
\author[1]{Binlin Dai}%
\affil[1]{Shanghai University of Finance and Economics}%


\vspace{-1em}



  
  \date{May 29, 2024}


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\begin{abstract}
We investigate Schwarz lemma in the framework of bicomplex numbers,
which are pairs of complex numbers making up a commutative ring with
zero-divisors. The bicomplex is a generalization of complex which has
closed relation with Frac- tal geometry, Minkowski Space-Time, Maxwell's
equations, Schr\selectlanguage{ngerman}ödinger equation and Gaussian pulse wave. In this paper,
we first construct a type of bicomplex Möbius transformation and obtain
some results : the mapping properties on bicom- plex spheres and
bicomplex ball, preserving the inverse points with respect to the
bicomplex sphere (0,1). Then we obtain the Poisson integral formula in
bicom- plex setting, and by using the Poisson integral formula, we give
the Schwarz lemma for bicomplex holomorphic functions in bicomplex
setting. Finally, we shall give the Schwarz lemma and the Schwarz-Pick
type lemma for holomorphic functions in bicomplex analysis. These
results may give new energy for the development of quan- tum mechanics.%
\end{abstract}\selectlanguage{ngerman}%



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