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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
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Abstract

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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