Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-19T22:17:44.973Z Has data issue: false hasContentIssue false
  • English
  • Français

The higher derivations of functions bounded in various senses

Published online by Cambridge University Press:  17 April 2009

Shinji Yamashita
Shinji Yamashita
Affiliation:
Department of Mathematics, Tokyo Metropolitan University, Fukasawa, Setagaya, Tokyo 158, Japan
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An extension (Theorem 1) of Schwarz and Pick's lemma motivates us to study the analogues for functions which are bounded in the sense of Bloch, normal, or yoshida. A typical result is that, for a function f holomorphic in D = {|z| < 1} and Bloch, that is, , with the expansion f(w) = c0 + cn (w − z)n + … (n ≧ 1) about 2 ε D, we have (1 − |z|2)n|f(n) (z)|/n! ≦ Anα, where An is an absclute constant; the estimate is sharp.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 34 , Issue 3 , December 1986 , pp. 321 - 334
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Lappan, P. A., “The spherical derivative and normal functions”, Ann. Acad. Sci. Fenn. Ser. A.I. Math. 3 (1977), 301–310.CrossRefGoogle Scholar
[2]Lehto, O. and Virtanen, K. I., “Boundary behaviour and normal meromorphic functions”, Acta Math. 97 (1957), 47–65.CrossRefGoogle Scholar
[3]Pommerenke, C., “On Bloch functions”, J. London Math. Soc. (2), 2 (1970), 689–695.CrossRefGoogle Scholar
[4]Szász, O., “Ungleichheitsbeziehungen für die Ableitungen einer Potenzreihe, die eine im Einheitskreise beschränkte Funktion darstellt”, Math. Z. 8 (1920), 303–309.CrossRefGoogle Scholar
[5]Yamashita, S., “On normal meromorphic functions”, Math. Z. 141 (1975), 139–145.CrossRefGoogle Scholar
[6]Yosida, K., “On a class of meromorphic functions”, Proc. Phys.-Math. Soc. Jap. 16 (1934), 227–235; Corrigendum, Proc. Phys.-Math. Soc. Jap. 16 (1934), 413.Google Scholar