A formula for the Bloch norm of a C 1 -function on the unit ball of n

Miroslav Pavlović

Czechoslovak Mathematical Journal (2008)

  • Volume: 58, Issue: 4, page 1039-1043
  • ISSN: 0011-4642

Abstract

top
For a C 1 -function f on the unit ball 𝔹 n we define the Bloch norm by f 𝔅 = sup d ˜ f , where d ˜ f is the invariant derivative of f , and then show that f 𝔅 = sup z , w 𝔹 z w ( 1 - | z | 2 ) 1 / 2 ( 1 - | w | 2 ) 1 / 2 | f ( z ) - f ( w ) | | w - P w z - s w Q w z | .

How to cite

top

Pavlović, Miroslav. "A formula for the Bloch norm of a $C^1$-function on the unit ball of $\mathbb {C}^n$." Czechoslovak Mathematical Journal 58.4 (2008): 1039-1043. <http://eudml.org/doc/37883>.

@article{Pavlović2008,
abstract = {For a $C^1$-function $f$ on the unit ball $\mathbb \{B\} \subset \mathbb \{C\} ^n$ we define the Bloch norm by $\Vert f\Vert _\mathfrak \{B\}=\sup \Vert \tilde\{d\}f\Vert ,$ where $\tilde\{d\}f$ is the invariant derivative of $f,$ and then show that \[ \Vert f\Vert \_\mathfrak \{B\}= \sup \_\{z,w\in \{\mathbb \{B\}\} \atop z\ne w\} (1-|z|^2)^\{1/2\}(1-|w|^2)^\{1/2\}\frac\{|f(z)-f(w)|\}\{|w-P\_wz-s\_wQ\_wz|\}.\]},
author = {Pavlović, Miroslav},
journal = {Czechoslovak Mathematical Journal},
keywords = {Bloch norm; Möbius transformation; Bloch norm; Möbius transformation},
language = {eng},
number = {4},
pages = {1039-1043},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A formula for the Bloch norm of a $C^1$-function on the unit ball of $\mathbb \{C\}^n$},
url = {http://eudml.org/doc/37883},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Pavlović, Miroslav
TI - A formula for the Bloch norm of a $C^1$-function on the unit ball of $\mathbb {C}^n$
JO - Czechoslovak Mathematical Journal
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 4
SP - 1039
EP - 1043
AB - For a $C^1$-function $f$ on the unit ball $\mathbb {B} \subset \mathbb {C} ^n$ we define the Bloch norm by $\Vert f\Vert _\mathfrak {B}=\sup \Vert \tilde{d}f\Vert ,$ where $\tilde{d}f$ is the invariant derivative of $f,$ and then show that \[ \Vert f\Vert _\mathfrak {B}= \sup _{z,w\in {\mathbb {B}} \atop z\ne w} (1-|z|^2)^{1/2}(1-|w|^2)^{1/2}\frac{|f(z)-f(w)|}{|w-P_wz-s_wQ_wz|}.\]
LA - eng
KW - Bloch norm; Möbius transformation; Bloch norm; Möbius transformation
UR - http://eudml.org/doc/37883
ER -

References

top
  1. Holland, F., Walsh, D., 10.1007/BF01451410, Math. Ann. 273 (1986), 317-335. (1986) Zbl0561.30025MR0817885DOI10.1007/BF01451410
  2. Nowak, M., 10.1080/17476930108815339, Complex Variables Theory Appl. 44 (2001), 1-12. (2001) MR1826712DOI10.1080/17476930108815339
  3. Pavlovi'c, M., On the Holland-Walsh characterization of Bloch functions, Proc. Edinb. Math. Soc. 51 (2008), 439-441. (2008) MR2465917
  4. Rudin, W., Function Theory in the Unit Ball of C n , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 241, Springer-Verlag, New York (1980). (1980) MR0601594

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.