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On a Conjecture of Clunie and Sheil‐Small

On a Conjecture of Clunie and Sheil‐Small R. R. HALL In this note I prove the following result which was conjectured by Clunie and Sheil-Small. THEOREM. Let fbea normalized convex function on A = {z : \z\ < 1} and w Let w - f(z) Then there exists a constant A, independent off and w, such that \c \ ^ A. I should like to thank Dr. Sheil-Small for mentioning this problem to me. It remains to find the best possible value of A: the present method depends on integral mean estimates and is unlikely to give this. I also give an integral formula which exactly represents a dense subclass of the normalized convex functions. LEMMA. Let f be a normalized convex function. Then for \z\ = r < 1, we have > -T. ^arc sin r Moreover this is sharp for each fixed r: the extremal function satisfies We remark that in the larger class of functions starlike of order {, the sharp lower 2 2 1 2 bound for the left hand side is ^/( l - r ): the extremal function is z(l — 2rz + z )' ' . For normalized univalent functions the sharp lower bound is 1 — r : http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

On a Conjecture of Clunie and Sheil‐Small

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/12.1.25
Publisher site
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Abstract

R. R. HALL In this note I prove the following result which was conjectured by Clunie and Sheil-Small. THEOREM. Let fbea normalized convex function on A = {z : \z\ < 1} and w Let w - f(z) Then there exists a constant A, independent off and w, such that \c \ ^ A. I should like to thank Dr. Sheil-Small for mentioning this problem to me. It remains to find the best possible value of A: the present method depends on integral mean estimates and is unlikely to give this. I also give an integral formula which exactly represents a dense subclass of the normalized convex functions. LEMMA. Let f be a normalized convex function. Then for \z\ = r < 1, we have > -T. ^arc sin r Moreover this is sharp for each fixed r: the extremal function satisfies We remark that in the larger class of functions starlike of order {, the sharp lower 2 2 1 2 bound for the left hand side is ^/( l - r ): the extremal function is z(l — 2rz + z )' ' . For normalized univalent functions the sharp lower bound is 1 — r :

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jan 1, 1980

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