# zbMATH — the first resource for mathematics

##### Examples
 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.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 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 ${ℂ}^{n}$. (English) Zbl 1243.32002

Let ${𝔹}_{n}:=\left\{\left({z}_{1},\cdots ,{z}_{n}\right)\in {ℂ}^{n}:{\sum }_{k=1}^{n}|{z}_{k}{|}^{2}<1\right\}$ be the unit ball in ${ℂ}^{n}$, ${𝔹}_{1}$ be the unit disc in $ℂ$. It is well known, that for any holomorphic function $f:{𝔹}_{1}\to {𝔹}_{1}$ the following Schwarz inequality holds

$|{f}^{\text{'}}\left(z\right)|⩽\frac{1-{|f\left(z\right)|}^{2}}{1-{|z|}^{2}},\phantom{\rule{1.em}{0ex}}z\in {𝔹}_{1}·$

There are many generalizations of this result. One of them is the following one due to S. Y. Dai and 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 $f:{𝔹}_{1}\to {𝔹}_{1}$ is holomorphic, then

$|{f}^{\left(m\right)}\left(z\right)|⩽\frac{m!\left(1-|f\left(z\right){|}^{2}\right)}{{\left(1-|z|}^{2}{\right)}^{m}}{\left(1+|z|\right)}^{m-1},\phantom{\rule{1.em}{0ex}}m\in ℕ,\phantom{\rule{4pt}{0ex}}z\in {𝔹}_{1}·$

The authors generalize this result on several complex variables as follows. Let $f:{𝔹}_{n}\to {𝔹}_{1}$ be holomorphic, $n\in ℕ$. Then for any multiindex $m=\left({m}_{1},\cdots ,{m}_{n}\right)\in {\left({ℤ}_{+}^{n}\right)}_{*}$

$|{\partial }^{m}f\left(z\right)|⩽{\left(\genfrac{}{}{0pt}{}{n+|m|-1}{n-1}\right)}^{n+2}{n}^{\frac{|m|}{2}}\frac{|m|!\left(1-|f\left(z\right){|}^{2}\right)}{{\left(1-|z|}^{2}{\right)}^{|m|}}{\left(1+|z|\right)}^{|m|-1},\phantom{\rule{1.em}{0ex}}z\in {𝔹}_{n},$

where ${\partial }^{m}f:=\frac{{\partial }^{|m|}f}{\partial {z}_{1}^{{m}_{1}}\cdots \partial {z}_{n}^{{m}_{n}}}$, $|m|={\sum }_{k=1}^{n}{m}_{k}$.

##### MSC:
 32A10 Holomorphic functions (several variables) 32A30 Generalizations of function theory to several variables 30C80 Maximum principle; Schwarz’s lemma, Lindelöf principle, etc. (one complex variable) 30H05 Bounded analytic functions 32A05 Power series, series of functions (several complex variables)
##### References:
 [1] Ruscheweyh, St.: Two remarks on bounded analytic functions. Serdica, 11, 200–202 (1985) [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) [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) [8] Maccluer, B., Stroethoff, K., Zhao, R.: Schwarz-Pick type estimates. Complex Var. Theory Appl., 48, 711–730 (2003) [9] Rudin, W.: Function Theory in the Unit Ball of $ℂ$n, Springer, New York, 1980