Necessary conditions for the existence of higher order extensions of univalent mappings from the disk to the ball

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:390

Article

Muir, Jerry R., Jr.

关键词: Biholomorphic mappings;    Extension operators;    Convex and starlike mappings;    Loewner chains;    Bloch functions;   

DOI  :  10.1016/j.jmaa.2012.01.035

来源: Elsevier

【 摘 要 】

If G : Cn-1 -> C is a holomorphic function such that G(0) = 0 and DG(0) = 0 and f is a normalized univalent mapping of the unit disk D subset of C, we consider the normalized extension of f to the Euclidean unit ball B subset of C-n given by Phi(G)(f)(z) = (f (z(1)) G(root f'(z(1))(z) over cap, z is an element of B, (z) over cap = (z(2), ..., z(n)). While for a given f, Phi(G)(f) will maintain certain geometric properties of f, such as convexity or starlikeness, if G is a polynomial of degree 2 of sufficiently small norm, these properties may be lost whenever G contains a nonzero term of higher degree. By establishing separate necessary and sufficient conditions for the extension of Loewner chains from D to B through Phi(G), we are able to completely classify those starlike and convex mappings f on D for which there exists a G with nonzero higher degree terms such that Phi(G) (f) is a mapping of the same type on B. (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

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