On coefficient estimates for a class of holomorphic mappings. (English) Zbl 1195.32011
The authors obtain bounds on the Taylor coefficients for starlike, normalized, biholomorphic mappings \(f\) from the unit ball \(B\) of a complex Banach space \(X\) into \(X\) such that \(f(x)-x\) has a multiple zero at \(x=0\) and \((Df(x))^{-1}f(x)\) belongs to a certain class \(M_g\) of holomorphic mappings \(B\to X\). The class is defined by means of a univalent function \(g\) on the unit disk \(\mathbb D\subset C\) with positive real part; when \(X={\mathbb C}^n\) and \(B={\mathbb D}^n\), \(M_g\) consists of holomorphic mappings \(p\) such that \(p(0)=0\), \(Dp(0)=I\), and \(p_j(z)/z_j\in g(D)\) for all \(z\in B\setminus\{0\}\).
The result generalizes bounds from [I. Graham, H. Hamada and G. Kohr, Can. J. Math. 54, No. 2, 324–351 (2002; Zbl 1004.32007); T. Liu and X. Liu, Sci. China, Ser. A 48, No. 7, 865–879 (2005; Zbl 1105.32013); H. Hamada, T. Honda and G. Kohr, J. Math. Anal. Appl. 317, No. 1, 302–319 (2006; Zbl 1092.32011)].
The result generalizes bounds from [I. Graham, H. Hamada and G. Kohr, Can. J. Math. 54, No. 2, 324–351 (2002; Zbl 1004.32007); T. Liu and X. Liu, Sci. China, Ser. A 48, No. 7, 865–879 (2005; Zbl 1105.32013); H. Hamada, T. Honda and G. Kohr, J. Math. Anal. Appl. 317, No. 1, 302–319 (2006; Zbl 1092.32011)].
Reviewer: Alexandr Yu. Rashkovsky (Stavanger)
MSC:
| 32H02 | Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables |
| 32A30 | Other generalizations of function theory of one complex variable |
| 30C45 | Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) |
Keywords:
holomorphic mappings on Banach spaces; starlike mappings; functions with positive real partReferences:
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