Tangential asymptotic values of bounded analytic functions

Satyanarayana, U. V.; Weiss, Max L.

doi:10.1090/S0002-9939-1973-0318498-X

Tangential asymptotic values of bounded analytic functions
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by U. V. Satyanarayana and Max L. Weiss PDF

Proc. Amer. Math. Soc. 41 (1973), 167-172 Request permission

Abstract:

Suppose $f$ is a bounded analytic function on the unit disc whose Fatou boundary function is approximately continuous from above at 1 with value 0. It is well known that $f$ tends to zero radially and therefore along every nontangential arc. Tanaka [3] and Boehme and Weiss [1] have shown that $f$ must also tend to zero along certain arcs which are tangential from above. The purpose of this paper is to improve their results by producing a larger collection of such tangential arcs along which $f$ tends to zero. We construct a class of examples to show that our result is actually better.

References

T. K. Boehme and Max L. Weiss, Extensions of Fatou’s theorem to tangential asymptotic values, Proc. Amer. Math. Soc. 27 (1971), 289–298. MR 273039, DOI 10.1090/S0002-9939-1971-0273039-9
U. V. Satyanarayana and Max L. Weiss, The geometry of convex curves tending to $1$ in the unit disc, Proc. Amer. Math. Soc. 41 (1973), 159–166. MR 318497, DOI 10.1090/S0002-9939-1973-0318497-8
Chuji Tanaka, On the metric cluster values of the bounded regular function in the unit disk, Mem. School Sci. Engrg. Waseda Univ. 31 (1967), 119–129. MR 247101