## A distortion theorem for biholomorphic mappings in $\textbf {C}^ 2$

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- by Roger W. Barnard, Carl H. FitzGerald and Sheng Gong PDF
- Trans. Amer. Math. Soc.
**344**(1994), 907-924 Request permission

## Abstract:

Let ${J_f}$ be the Jacobian of a normalized biholomorphic mapping*f*from the unit ball ${B^2}$ into ${\mathbb {C}^2}$. An expression for the $\log \det {J_f}$ is determined by considering the series expansion for the renormalized mappings

*F*obtained from

*f*under the group of holomorphic automorphisms of ${B^2}$. This expression is used to determine a bound for $|\det {J_f}|$ and $|\arg \det {J_f}|$ for

*f*in a compact family

*X*of normalized biholomorphic mappings from ${B^2}$ into ${\mathbb {C}^2}$ in terms of a bound $C(X)$ of a certain combination of second-order coefficients. Estimates are found for $C(X)$ for the specific family

*X*of normalized convex mappings from ${B^2}$ into ${\mathbb {C}^2}$.

## References

- Roger W. Barnard, Carl H. FitzGerald, and Sheng Gong,
*The growth and $1/4$-theorems for starlike mappings in $\textbf {C}^n$*, Pacific J. Math.**150**(1991), no. 1, 13–22. MR**1120709**
H. Cartan, - Peter L. Duren,
*Univalent functions*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR**708494** - Peter Duren and Walter Rudin,
*Distortion in several variables*, Complex Variables Theory Appl.**5**(1986), no. 2-4, 323–326. MR**846500**, DOI 10.1080/17476938608814152 - Sheng Gong,
*A remark on the Möbius transformations. I*, Chuncui Shuxue yu Yingyong Shuxue**1**(1985), 1–15 (Chinese, with English summary). MR**874226** - Keizo Kikuchi,
*Starlike and convex mappings in several complex variables*, Pacific J. Math.**44**(1973), 569–580. MR**322210** - Walter Rudin,
*Function theory in the unit ball of $\textbf {C}^{n}$*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR**601594** - T. J. Suffridge,
*The principle of subordination applied to functions of several variables*, Pacific J. Math.**33**(1970), 241–248. MR**261040** - T. J. Suffridge,
*Starlikeness, convexity and other geometric properties of holomorphic maps in higher dimensions*, Complex analysis (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976) Lecture Notes in Math., Vol. 599, Springer, Berlin, 1977, pp. 146–159. MR**0450601**

*Sur la possibilité d’étendre sux fonctions de plusieurs variables complexes la théorie des fonctions univalentes*, Note added to P. Montel,

*Leçons sur les fonctions univalentes on multivalentes*, Gauthier-Villars, Paris, 1993, pp. 129-155.

## Additional Information

- © Copyright 1994 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**344**(1994), 907-924 - MSC: Primary 32H02
- DOI: https://doi.org/10.1090/S0002-9947-1994-1250815-7
- MathSciNet review: 1250815