zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Schwarz-Pick estimates for bounded holomorphic functions in the unit ball of $\Bbb C^n$. (English) Zbl 1243.32002
Let $\Bbb B_n:=\{(z_1,\dots,z_n)\in\Bbb C^n:\sum_{k=1}^n|z_k|^2<1\}$ be the unit ball in $\Bbb C^n$, $\Bbb B_1$ be the unit disc in $\Bbb C$. It is well known, that for any holomorphic function $f:\Bbb B_1\rightarrow\Bbb B_1$ the following Schwarz inequality holds $$ |f'(z)|\leqslant\frac{1-|f(z)|^2}{1-|z|^2},\quad z\in\Bbb B_1. $$ There are many generalizations of this result. One of them is the following one due to {\it S. Y. Dai} and {\it Y. F. Pan} [Proc. Am. Math. Soc. 136, No. 2, 635--640 (2008; Zbl 1133.30005)], which gives the estimate of higher order derivatives: if $f:\Bbb B_1\rightarrow\Bbb B_1$ is holomorphic, then $$ |f^{(m)}(z)|\leqslant\frac{m!(1-|f(z)|^2)}{(1-|z|^2)^m}(1+|z|)^{m-1},\quad m\in\Bbb N,\ z\in\Bbb B_1. $$ The authors generalize this result on several complex variables as follows. Let $f:\Bbb B_n\rightarrow\Bbb B_1$ be holomorphic, $n\in\Bbb N$. Then for any multiindex $m=(m_1,\dots,m_n)\in(\Bbb Z_+^n)_*$ $$ |\partial^mf(z)|\leqslant\binom{n+|m|-1}{n-1}^{n+2}n^{\frac{|m|}{2}}\frac{|m|!(1-|f(z)|^2)}{(1-|z|^2)^{|m|}}(1+|z|)^{|m|-1},\quad z\in\Bbb B_n, $$ where $\partial^mf:=\frac{\partial^{|m|}f}{\partial z_1^{m_1}\dots\partial z_n^{m_n}}$, $|m|=\sum_{k=1}^nm_k$.

32A10Holomorphic functions (several variables)
32A30Generalizations of function theory to several variables
30C80Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable)
30H05Bounded analytic functions
32A05Power series, series of functions (several complex variables)
Full Text: DOI
[1] Ruscheweyh, St.: Two remarks on bounded analytic functions. Serdica, 11, 200--202 (1985) · Zbl 0581.30009
[2] Anderson, J. M., Rovnyak, J.: On generalized Schwarz-Pick estimates. Mathematika, 53, 161--168 (2006) · Zbl 1120.30001 · doi:10.1112/S0025579300000085
[3] Avkhadiev, F. G., Wirths, K. J.: Schwarz-Pick inequalities for derivatives of arbitrary order. Constr. Approx., 19, 265--277 (2003) · Zbl 1018.30018 · doi:10.1007/s00365-002-0503-4
[4] Beneteau, C., Dahlner, A., Khavinson, D.: Remarks on the Bohr phenomenon. Comput. Methods Funct. Theory, 4, 1--19 (2004) · Zbl 1067.30094
[5] Maccluer, B., Stroethoff, K., Zhao, R.: Generalized Schwarz-Pick estimates. Proc. Amer. Math. Soc., 131, 593--599 (2002) · Zbl 1012.30015 · doi:10.1090/S0002-9939-02-06588-7
[6] Dai, S. Y., Pan, Y. F.: Note on Schwarz-Pick estimates for Bounded and Positive Real Part Analytic Functions. Proc. Amer. Math. Soc., 136, 635--640 (2008) · Zbl 1133.30005 · doi:10.1090/S0002-9939-07-09064-8
[7] Anderson, J. M., Dritschel, M. A., Rovnyak, J.: Schwarz-Pick inequalities for the Schur-Agler class on the polydisk and unit ball. Computational Methods and Function Theory, 8, 339--362 (2008) · Zbl 1157.30019
[8] Maccluer, B., Stroethoff, K., Zhao, R.: Schwarz-Pick type estimates. Complex Var. Theory Appl., 48, 711--730 (2003) · Zbl 1031.30012
[9] Rudin, W.: Function Theory in the Unit Ball of $\mathbb{C}$n, Springer, New York, 1980 · Zbl 0495.32001