×

Some refined Schwarz-Pick lemmas. (English) Zbl 1026.32023

The author studies some Schwarz-type theorems for holomorphic mappings \(f\:\mathbb B_n\rightarrow\mathbb B_m\), where \(\mathbb B_k\) denotes the unit Euclidean balls in \(\mathbb C^k\). The main result of the paper is the following theorem. If \(f(a)=b\) and \(f(A)=B\), then \[ \beta_m\biggl(Sf^\ast(a)\frac{\varphi_a(A)}{|\varphi_a(A)|},\;Tf^\ast(A)\frac{\varphi_A(a)}{|\varphi_A(a)|}\biggr)\leq 2\beta_n(a,A), \] where \(\beta_k\) denotes the Bergman distance for \(\mathbb B_k\), \(\varphi_c\:\mathbb B_n\rightarrow\mathbb B_n\) is the Möbius automorphism such that \(\varphi(0)=c\), \(\varphi(c)=0\), \(S\) and \(T\) are unitary isomorphisms of \(\mathbb C^m\) such that \(S(\varphi_b(B))=T(\varphi_B(b))\), and \(f^\ast(c):=\varphi_{f(c)}'(f(c))f'(c)\varphi_c'(0)\).

MSC:

32F45 Invariant metrics and pseudodistances in several complex variables
30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. F. Beardon, The Schwarz–Pick lemma for derivatives, Proc. Amer. Math. Soc. 125 (1997), 3255–3256. · Zbl 0883.30018 · doi:10.1090/S0002-9939-97-03906-3
[2] A. F. Beardon and T. K. Carne, A strengthening of the Schwarz–Pick inequality, Amer. Math. Monthly 99 (1992), 216–217. · Zbl 0753.30015 · doi:10.2307/2325054
[3] D. M. Burns and S. G. Krantz, Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary, J. Amer. Math. Soc. 7 (1994), 661–676. JSTOR: · Zbl 0807.32008 · doi:10.2307/2152787
[4] C. C. Cowen and B. D. MacCluer, Composition operators on spaces of analytic functions, Stud. Adv. Math., CRC Press, Boca Raton, FL, 1995. · Zbl 0873.47017
[5] P. L. Duren, Univalent functions, Grundlehren Math. Wiss., 259, Springer, New York, 1983.
[6] J. B. Garnett, Bounded analytic functions, Pure Appl. Math., 96, Academic Press, New York, 1981. · Zbl 0469.30024
[7] P. R. Mercer, Sharpened versions of the Schwarz lemma, J. Math. Anal. Appl. 205 (1997), 509–511. · Zbl 0871.30023 · doi:10.1006/jmaa.1997.5217
[8] R. Osserman, A new variant of the Schwarz–Pick–Ahlfors lemma, Manuscripta Math. 100 (1999), 123–129. · Zbl 0938.30017 · doi:10.1007/s002290050231
[9] ——, A sharp Schwarz inequality on the boundary, Proc. Amer. Math. Soc. 128 (2000), 3513–3517. · Zbl 0963.30014 · doi:10.1090/S0002-9939-00-05463-0
[10] W. Rudin, Function theory in the unit ball of \(\mathbb C^n,\) Grundlehren Math. Wiss., 241, Springer, New York, 1980. · Zbl 0495.32001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.