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A Landau type theorem for holomorphic mappings of bounded convex balanced domains in \({\mathbb{C}}^ n\). (English) Zbl 0722.32005

For \(\lambda\in (0,1]\), let \(F_{\lambda}=\{f\in H(U):\) \(f(0)=0\), \(f'(0)=\lambda\), and f(z)\(\in U\) for all \(z\in U\}\) and, for each \(f\in F_{\lambda}\), let \(\rho (f)=\sup \{\rho >0:\) for some domain \(\Omega\), \(0\in \Omega \subset U\), f maps \(\Omega\) univalently onto \(\rho\Omega\}\) (where U is the open unit disc). Landau’s constant \(L_{\lambda}\) is defined to be inf\(\{\rho\) (f): \(f\in F_{\lambda}\}\), and it is known that \(L_{\lambda}=(\lambda /(1+\sqrt{1-\lambda^ 2}))^ 2.\) In Kodai Math. J. 9, 241-244 (1986; Zbl 0604.32019)] the author generalized this result, with U replaced by the unit ball \(B_ n\subset {\mathbb{C}}^ n\). Here the extends this result to bounded, convex, balanced subsets of \({\mathbb{C}}^ n\). This result is obtained by examining domains of univalence of certain holomorphic mappings between bounded, convex, balanced domains, which follows in turn from consideration of the Carathéodory and Kobayashi metrics.
Reviewer: R.M.Aron (Kent)

MSC:

32A30 Other generalizations of function theory of one complex variable
30D50 Blaschke products, etc. (MSC2000)
32H30 Value distribution theory in higher dimensions
30C55 General theory of univalent and multivalent functions of one complex variable
32F45 Invariant metrics and pseudodistances in several complex variables

Citations:

Zbl 0604.32019
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