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On the Zeros of Functions with Derivatives in H1 and H∞

Published online by Cambridge University Press:  20 November 2018

James Wells
James Wells*
Affiliation:
Texas Tech University, Lubbock, Texas University of Kentucky, Lexington, Kentucky
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Let {zk},0 < |zk| < 1, be a given sequence of points in the open unit disc D = {z: |z| < 1} and let E be its set of limit points on the unit circle T. In this note we consider the problem of finding conditions on the sequence {zk} which will ensure the existence of a function f analytic in D satisfying

(A)

and whose derivative f′ belongs to the Hardy class H1 or, alternatively, whose derivatives of all orders are bounded in D. We shall prove the following two theorems.

THEOREM 1. If

(1)

(2)

and

(3)

then there is a function f analytic in D which satisfies (A) and its derivative f′ belongs to H1.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 2 , 01 April 1970 , pp. 342 - 347
Copyright
Copyright © Canadian Mathematical Society 1970

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