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The high order Schwarz-Pick lemma on complex Hilbert balls. (English) Zbl 1211.32001

The aim of the article is to establish a high order Schwarz lemma for holomorphic mappings between the unit balls 𝔹 and 𝔹 ˜ in complex Hilbert spaces X and Y, respectively. For z𝔹 and βX, we denote by H z (β,β) the quantity

H z (β,β):=(1-z 2 )β 2 +|β,z| 2 (1-z 2 ) 2 ·

For a holomorphic mapping f:𝔹𝔹 ˜, we denote by D k f(z) the Fréchet derivative of order k at a point z𝔹 and by D k f(z)·(β 1 ,,β k ) its evaluation at a k-tuple (β 1 ,,β k ) of vectors in X.

Then the authors’ first main result is as follows:

Theorem 1: Let f:𝔹𝔹 ˜ be a holomorphic mapping. Then, for any positive integer k, any z𝔹 and βX{0}, one has

H f(z) (D k f(z)·β k ,D k f(z)·β k )(k!) 2 p(z,β) 2(k-1) (H z (β,β)) k ,

where β k =(β,,β) and

p(z,β):=1+|β,z| ((1-z 2 )β 2 +|β,z| 2 ) 1/2 ·

With the same method of proof as for Theorem 1, the authors obtain

Theorem 2: For holomorphic mappings f:X with a positive real part, one has

|D k f(z)·β k |2k!Ref(z)}p(z,β) k-1 H z (β,β) k/2

for z𝔹, an integer k1, and any βX{0}.

Also, the norm of the k-th order Fréchet derivative is estimated in the following two theorems, namely:

Theorem 3: Let f:𝔹𝔹 ˜ be a holomorphic mapping. Then for any positive integer k and any z𝔹, one has

D k f(z)k k 1-f(z) 2 (1+z) k-1 (1-z 2 ) k ·

Finally, they obtain for functions as in Theorem 2:

Theorem 4: For holomorphic mappings f:𝔹 with a positive real part one has, with integer valued k1, for z𝔹 the estimate

D k f(z)2k k {Ref(z)}(1+z) k-1 (1-z 2 ) k ·

MSC:
32A10Holomorphic functions (several variables)
32F45Invariant metrics and pseudodistances
References:
[1]Abraham R, Marsden J E, Ratiu T. Manifolds, Tensor Analysis, and Applications, 2nd ed. New York: Springer-Verlag, 1988, 40–116
[2]Chen Z H, Liu Y. Schwarz-Pick estimates for bounded holomorphic functions in the unit ball of n. Acta Math Sinica Engl Ser, 2010, 5: 901–908 · Zbl 1243.32002 · doi:10.1007/s10114-010-7487-y
[3]Dai S Y, Chen H H, Pan Y F. The Schwarz-Pick lemma of high order in several variables. Preprint
[4]Dai S Y, Pan Y F. Note on Schwarz-Pick estimates for bounded and positive real part analytic functions, Proc Amer Math Soc, 2008, 136: 635–640 · Zbl 1133.30005 · doi:10.1090/S0002-9939-07-09064-8
[5]Maccluer B, Stroethoff K, Zhao R H. Generalized Schwarz-Pick estimates. Proc Amer Math Soc, 2003, 131: 593–599 · Zbl 1012.30015 · doi:10.1090/S0002-9939-02-06588-7
[6]Renaud A. Quelques propriétés des applications analytiques d’une boule de dimension infinie dans une autre. Bull Sci Math, 1973, 97: 129–159
[7]Zhang M Z. Generalized Schwarz-Pick lemma. Acta Math Sinica Chin Ser, 2006, 49: 613–616