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Angular and Tangential Limits of Blaschke Products and their Successive Derivatives

Published online by Cambridge University Press:  20 November 2018

G. T. Cargo
G. T. Cargo*
Affiliation:
Syracuse University
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In this paper, we shall be concerned with bounded, holomorphic functions of the form

where

(1)

(2)

and

(3)

B(z{an}) is called a Blaschke product, and any sequence {an} which satisfies (2) and (3) is called a Blaschke sequence. For a general discussion of the properties of Blaschke products, see (18, pp. 271-285) or (14, pp. 49-52).

According to a theorem due to Riesz (15), a Blaschke product has radial limits of modulus one almost everywhere on C = {z: |z| = 1}. Moreover, it is common knowledge that, if a Blaschke product has a radial limit at a point, then it also has an angular limit at the point (see 14, p. 19 and 6, p. 457).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 14 , 1962 , pp. 334 - 348
Copyright
Copyright © Canadian Mathematical Society 1962

References

1. Bagemihl, F. and Seidel, W., A general principle involving Baire category, with applications to function theory and other fields, Proc. Nat. Acad. Sci. U.S.A., 39 (1953), 1068–1075.Google Scholar
2. Cargo, G. T., The radial images of Blaschke products, J. London Math. Soc, to appear.Google Scholar
3. Carleson, L., On a class of meromorphic functions and its associated exceptional sets (thesis, Uppsala, 1950).Google Scholar
4. Chaundy, T., The differential calculus (Oxford, 1953).Google Scholar
5. Collingwood, E. F., On sets of maximum indétermination of analytic functions, Math. Z., 67 (1957), 377–396.Google Scholar
6. Dienes, P., The Taylor series (Oxford, 1931).Google Scholar
7. Frostman, O., Potentiel d'équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions (thèse, Lund, 1935).Google Scholar
8. Frostman, O., Sur les produits de Blaschke, Kungl. Fysiogr. Sâllsk. i Lund Fôrh., Bd. 12, Nr. 15 (1942), 169–182.Google Scholar
9. Kinney, J. R., Boundary behavior of Blaschke products in the unit circle, Proc. Amer. Math. Soc, 12 (1961), 484–488.Google Scholar
10. Knopp, K., Theory and application of infinite series (2nd English éd.; New York, 1947).Google Scholar
11. Lohwater, A. J. and Piranian, G., The boundary behavior of functions analytic in a disk, Ann. Acad. Sci. Fenn. Ser. A. I., 239 (1957).Google Scholar
12. Munroe, M. E., Introduction to measure and integration (Cambridge, Mass., 1953).Google Scholar
13. Osgood, W. F., Lehrbuch der Funktionentheorie, vol. 1 (Berlin-Leipzig: fönfte Auflage, 1928).Google Scholar
14. Priwalow, I. I., Randeigenschaften analytischer Funktionen (Berlin, 1956).Google Scholar
15. Riesz, F., Ueber die Randwerte einer analytischen Funktion, Math. Z., 18 (1923), 87–95.Google Scholar
16. Saks, S. and Zygmund, A., Analytic functions (Warsaw, 1952).Google Scholar
17. Zygmund, A., On a theorem of Littlewood, Summa Brasil. Math. (1949), Fasc. 5.Google Scholar
18. Zygmund, A., Trigonometric series, vol. 1 (2nd ed.; Cambridge, 1959).Google Scholar