The geometry of convex curves tending to in the unit disc

Authors:
U. V. Satyanarayana and Max L. Weiss

Journal:
Proc. Amer. Math. Soc. **41** (1973), 159-166

MSC:
Primary 30A78

DOI:
https://doi.org/10.1090/S0002-9939-1973-0318497-8

MathSciNet review:
0318497

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Abstract: Let be a real convex function defined near otherwise. The family of curves, , in the unit disc is investigated. The union of the closures in the maximal ideal space of of these curves is seen to be a union of nontrivial parts and each such part is hit by the closure of each such curve. Using results of Hoffman and the Wermer map onto such a part, nets in the disc tending to points of the part are located on the curves as explicitly as can be expected. One striking consequence of this coordinatization is the fact that the closure of any such curve in a part is invariably the image under the Wermer map of an oricycle. This is not true for the limiting case of the family of Stolz rays at 1. Other detailed results of a similar nature are derived.

**[1]**T. K. Boehme and Max L. Weiss,*One-sided boundary behavior for certain harmonic functions*, Proc. Amer. Math. Soc.**27**(1971), 280-288. MR**44**#5491. MR**0288293 (44:5491)****[2]**K. Hoffman,*Bounded analytic functions and Gleason parts*, Ann. of Math. (2)**86**(1967), 74-111. MR**35**#5945. MR**0215102 (35:5945)****[3]**M. Rosenfeld and Max L. Weiss,*Bounded analytic functions tending radially to zero*, Proc. London Math. Soc. (3)**18**(1968), 714-726. MR**38**#1269. MR**0232946 (38:1269)****[4]**M. L. Weiss,*Note on a theorem of Beurling*, Proc. Amer. Math. Soc.**19**(1968), 793-795. MR**37**#3006. MR**0227422 (37:3006)****[5]**Dennis H. Wortman,*Interpolating sequences in convex regions of the open unit disc*, Doctoral Dissertation, M.I.T , Cambridge, Mass., 1972.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1973-0318497-8

Keywords:
Convex curve,
Wermer map,
part,

Article copyright:
© Copyright 1973
American Mathematical Society