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On the Zeros of Functions with Derivatives in H 1 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 H 1 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 H 1.

Information

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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