Geodesic
On the Zeros of Functions with Derivatives in H 1 and H ∞
Canadian journal of mathematics, Tome 22 (1970) no. 2, pp. 342-347

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Wells, James. On the Zeros of Functions with Derivatives in H 1 and H ∞. Canadian journal of mathematics, Tome 22 (1970) no. 2, pp. 342-347. doi: 10.4153/CJM-1970-042-x
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[1] 1. Beurling, A., Ensembles exceptionnels, Acta Math. 72 (1940), 1–13. Google Scholar

[2] 2. Carleson, L., Sets of uniqueness for functions regular in the unit circle, Acta Math. 87 (1952), 325–345. Google Scholar

[3] 3. Carleson, L., On the zeros of functions with bounded Dirichlet integrals, Math. Z. 56 (1952), 289–295. Google Scholar

[4] 4. Caughran, James G., Analytic functions with Hv derivative, Thesis, University of Michigan, Ann Arbor, 1967. Google Scholar

[5] 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] 6. Hardy, G. H. and Littlewood, J. E., A convergence theorem for Fourier series, Math. Z. 28 (1928), 565–634. Google Scholar

[7] 7. Hardy, G. H. and Littlewood, J. E., Some properties of fractional integrals. II, Math. Z. 34 (1931), 403–439. Google Scholar

[8] 8. Hoffman, K., Banach spaces of analytic functions (Prentice-Hall, Englewood Cliffs, N.J., 1962). Google Scholar

[9] 9. Phillip, Novinger, Holomorphic functions with infinitely differentiate boundary values (to appear in Illinois J. Math.). Google Scholar

[10] 10. Walter, Rudin, Real and complex analysis (McGraw-Hill, New York, 1966). Google Scholar

[11] 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] 12. Taylor, B. A. and Williams, D. L., On closed ideals in Ax, Notices Amer. Math. Soc. 16 (1969), 144. Google Scholar

[13] 13. Zygmund, A., Trigonometric series, Vol. II, 2nd ed. (Cambridge Univ. Press, New York, 1959). Google Scholar

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