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Schwarz-Pick estimates for bounded holomorphic functions in the unit ball of n . (English) Zbl 1243.32002

Let 𝔹 n :={(z 1 ,,z n ) n : k=1 n |z k | 2 <1} be the unit ball in n , 𝔹 1 be the unit disc in . It is well known, that for any holomorphic function f:𝔹 1 𝔹 1 the following Schwarz inequality holds

|f ' (z)|1-|f(z)| 2 1-|z| 2 ,z𝔹 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 𝔹 1 is holomorphic, then

|f (m) (z)|m!(1-|f(z)| 2 ) (1-|z| 2 ) m (1+|z|) m-1 ,m,z𝔹 1 ·

The authors generalize this result on several complex variables as follows. Let f:𝔹 n 𝔹 1 be holomorphic, n. Then for any multiindex m=(m 1 ,,m n )( + n ) *

| m f(z)|n+|m|-1 n-1 n+2 n |m| 2 |m|!(1-|f(z)| 2 ) (1-|z| 2 ) |m| (1+|z|) |m|-1 ,z𝔹 n ,

where m f:= |m| f z 1 m 1 z n m n , |m|= k=1 n m k .


MSC:
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)
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 n, Springer, New York, 1980