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On Derivatives of the Maximum Modulus of a Starlike Function

On Derivatives of the Maximum Modulus of a Starlike Function ON DERIVATIVES OF THE MAXIMUM MODULUS OF A STARLIKE FUNCTION R. R. LONDO N 1. Introduction Let / be regular in the unit disc and define M{r) = max \f(z)\. \z\ = r By Hadamard's three circles theorem [3, p. 172] M(r) has a left and right derivative at every point of (0,1) . A result of Blumenthal [4, p. 22] shows that for any point p of (0,1) and for any suitably small positive e, M(r) has the form k/m £ <x (r — p) (oc real, m a positive integer) k k * = o on each of the intervals (p — e, p), {p,p + e). It follows that M{r) is analytic on M ( (p — e, p + e)\{p], and that M (p + 0) and M "\p — 0) exist as possibly infinite limits. Next let f(z) be regular in the unit disc and such that Then f(\z\ < 1) is starshaped with respect to the origin, and / is called starlike. Pommerenke [1] has proved that for some number a = a(/ ) e [0, 2] , called the order of / , hm —— = <x 1.1 r-i M(r) where M http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

On Derivatives of the Maximum Modulus of a Starlike Function

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References (2)

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/13.3.207
Publisher site
See Article on Publisher Site

Abstract

ON DERIVATIVES OF THE MAXIMUM MODULUS OF A STARLIKE FUNCTION R. R. LONDO N 1. Introduction Let / be regular in the unit disc and define M{r) = max \f(z)\. \z\ = r By Hadamard's three circles theorem [3, p. 172] M(r) has a left and right derivative at every point of (0,1) . A result of Blumenthal [4, p. 22] shows that for any point p of (0,1) and for any suitably small positive e, M(r) has the form k/m £ <x (r — p) (oc real, m a positive integer) k k * = o on each of the intervals (p — e, p), {p,p + e). It follows that M{r) is analytic on M ( (p — e, p + e)\{p], and that M (p + 0) and M "\p — 0) exist as possibly infinite limits. Next let f(z) be regular in the unit disc and such that Then f(\z\ < 1) is starshaped with respect to the origin, and / is called starlike. Pommerenke [1] has proved that for some number a = a(/ ) e [0, 2] , called the order of / , hm —— = <x 1.1 r-i M(r) where M

Journal

Bulletin of the London Mathematical Society – Wiley

Published: May 1, 1981

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