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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

References

1. Beurling, A., Ensembles exceptionnels, Acta Math. 72 (1940), 1–13.Google Scholar
2. Carleson, L., Sets of uniqueness for functions regular in the unit circle, Acta Math. 87 (1952), 325–345.Google Scholar
3. Carleson, L., On the zeros of functions with bounded Dirichlet integrals, Math. Z. 56 (1952), 289–295.Google Scholar
4. Caughran, James G., Analytic functions with Hv derivative, Thesis, University of Michigan, Ann Arbor, 1967.Google Scholar
5. Caughran, James G., Two results concerning the zeros of functions with finite Dirichlet integral, Can. J. Math. 21 (1969), 312–316.Google Scholar
6. Hardy, G. H. and Littlewood, J. E., A convergence theorem for Fourier series, Math. Z. 28 (1928), 565–634.Google Scholar
7. Hardy, G. H. and Littlewood, J. E., Some properties of fractional integrals. II, Math. Z. 34 (1931), 403–439.Google Scholar
8. Hoffman, K., Banach spaces of analytic functions (Prentice-Hall, Englewood Cliffs, N.J., 1962).Google Scholar
9. Phillip, Novinger, Holomorphic functions with infinitely differentiate boundary values (to appear in Illinois J. Math.).Google Scholar
10. Walter, Rudin, Real and complex analysis (McGraw-Hill, New York, 1966).Google Scholar
11. Shapiro, H. S. and Shields, A. L., On the zeros of functions with finite Dirichlet integral and some related function spaces, Math. Z. 80 (1962), 217–229.Google Scholar
12. Taylor, B. A. and Williams, D. L., On closed ideals in Ax, Notices Amer. Math. Soc. 16 (1969), 144.Google Scholar
13. Zygmund, A., Trigonometric series, Vol. II, 2nd ed. (Cambridge Univ. Press, New York, 1959).Google Scholar