Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-28T05:06:57.166Z Has data issue: false hasContentIssue false
  • English
  • Français

The length of ray-images under starlike mappings

Published online by Cambridge University Press:  26 February 2010

R. R. Hall
R. R. Hall
Affiliation:
The Department of Mathematics, The University of York, England
Get access

Extract

Let w = f(z) be regular and schlicht for |z|, and f (0) = 0.

Suppose that f maps the unit disc {z : |z| < 1} onto a domain D starlike with respect to w = 0. Let C(r, θ) be the image in D of the ray joining z = 0 to z = reiθ, and let

be its length. Sheil–Small [1] proved that l(r, θ) < (1 + log 4) | f (reiθ)|, and conjectured the following result, which it is my aim to prove in this paper.

Type
Research Article
Information
Mathematika , Volume 23 , Issue 2 , December 1976 , pp. 147 - 150
Copyright
Copyright © University College London 1976

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Sheil-Small, T.. “Some conformal mapping inequalities for starlike and convex functions”, J. London Math. Soc. (2), 1 (1969), 577–587.Google Scholar
2.Zygmund, A.. Trigonometric Series (Cambridge, 1968).Google Scholar