Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-11T03:21:32.960Z Has data issue: false hasContentIssue false
  • English
  • Français

Tangential Limits for Certain Classes of Analytic Functions

Published online by Cambridge University Press:  26 February 2010

J. B. Twomey
J. B. Twomey
Affiliation:
Department of Mathematics, University College, Cork, Ireland.
Get access

Extract

We begin by denning the notion of a tangential limit for a function f denned in the unit disc

Let be a positive continuous function on (0, 1) for which

Suppose B>0, -, and define

where The region . makes tangential contact with the boundary U of the unit disc at ei; when (r) = ( l - r2), for instance, (, , 1) is the disc with radius and centre ei

Type
Research Article
Information
Mathematika , Volume 36 , Issue 1 , June 1989 , pp. 39 - 49
Copyright
Copyright University College London 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Carleson, L.. On a Class of Meromorphic Functions and its Associated Exceptional Sets (Appelbergs Boktryckeri, Uppsala, 1950).Google Scholar
2.Collingwood, E. F. and Lohwater, A. J.. The Theory of Cluster Sets (Cambridge University Press, 1966).CrossRefGoogle Scholar
3.Fejer, L.. Trigonometrische reihen and potenzreihen mit mehrfach monotoner keoffizienfolge. Trans. Amer. Math. Soc., 39 (1936), 1859.CrossRefGoogle Scholar
4.Garnet, J. B.. Bounded Analytic Functions (Academic Press, New York, 1981).Google Scholar
5.Knopp, K.. Theory and Application of Infinite Series (Blackie and Son Ltd., London, 1951).Google Scholar
6.Littlewood, J. E.. On a theorem of Fatou. J. London Math. Soc., 2 (1927), 172176.CrossRefGoogle Scholar
7.Meyers, N. G.. A theory of capacities for potentials of functions in Lebesgue classes. Math. Scand., 26 (1970), 255292.CrossRefGoogle Scholar
8.Nagel, A.Ruden, W. and Shapiro, J. H.. Tangential boundary behavior of functions in Dirichlet-type spaces. Annals of Maths., 116 (1982), 331360.CrossRefGoogle Scholar
9.Nagel, A.Rudin, W. and Shapiro, J. H.. Tangential boundary behavior of harmonic extensions of Lp potentials. Conference of harmonic analysis (1981: Chicago, 111.) (Wadsworth Mathematical Series, 1983, Wadsworth, California), 533548.Google Scholar
10.Nagel, A. and Stein, E. M.. On certain maximal functions and approach regions. Advances in Mathematics, 54 (1984), 83106.CrossRefGoogle Scholar
11.Twomey, J. B.. On certain sums of Fourier-Stieltjes coefficients. Trans. Amer. Math. Soc., 280 (1983), 611621.Google Scholar
12.Twomey, J. B.. Tangential limits of starlike univalent functions. Proc. Amer. Math. Soc., 97 (1986), 4954.CrossRefGoogle Scholar
13.Twomey, J. B.. Tangential boundary behaviour of the Cauchy integral. J. London Math. Soc. (2), 37 (1988), 447454.Google Scholar
14.Zygmund, A.. Trigonometric Series, Vol. 1, 2nd rev. ed. (Cambridge Univ. Press, New York, 1959).Google Scholar