Estimates of the harmonic measure of a continuum in the unit disk

Authors:
Carl H. FitzGerald, Burton Rodin and Stefan E. Warschawski

Journal:
Trans. Amer. Math. Soc. **287** (1985), 681-685

MSC:
Primary 30C85; Secondary 31A15

DOI:
https://doi.org/10.1090/S0002-9947-1985-0768733-1

MathSciNet review:
768733

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The harmonic measure of a continuum in the unit disk is estimated from below in two ways. The first estimate is in terms of the angle subtended by the continuum as viewed from the origin. This result is a dual to the Milloux problem. The second estimate is in terms of the diameter of the continuum. This estimate was conjectured earlier as a strengthening of a theorem of D. Gaier. In preparation for the proofs several lemmas are developed. These lemmas describe some properties of the Riemann mapping function of a disk with radial incision onto a disk.

**[1]**L. V. Ahlfors,*Conformal invariants*:*topics in geometric function theory*, McGraw-Hill, New York, 1973. MR**0357743 (50:10211)****[2]**K. F. Barth, D. A. Brannan and W. K. Hayman,*Research problems in complex analysis*, Bull. London Math. Soc.**16**(1984), 490-517. MR**751823 (86b:30004)****[3]**D. Gaier,*Estimates of conformal mapping near the boundary*, Indiana Univ. Math. J.**21**(1972), 581-595. MR**0293072 (45:2151)****[4]**-,*A note on Hall's Lemma*, Proc. Amer. Math. Soc.**37**(1973), 97-99. MR**0310231 (46:9333)****[5]**W. K. Hayman,*On a theorem of Tord Hall*, Duke Math. J.**41**(1974), 25-26. MR**0335833 (49:611)****[6]**J. A. Jenkins,*On a lemma of Tord Hall*, Bull. Inst. Math. Acad. Sinica**2**(1979), 371-375. MR**0382627 (52:3509)****[7]**-,*On a problem concerning harmonic measure*, Math. Z.**135**(1974), 279-283. MR**0335787 (49:567)****[8]**M. Lavrientiev,*On the theory of conformal transformations*Trudy Mat. Inst. Steklov.**5**(1934), 159-245. (Russian)**[9]**L. Liao,*Certain extremal problems concerning module and harmonic measure*, J. Anal. Math.**40**(1981), 1-42. MR**659784 (83i:30022)****[10]**B. J. Maitland,*A note on functions regular and bounded in the unit circle and small at a set of points near the circumference of the circle*, Proc. Cambridge Philos. Soc.**35**(1939), 382-388. MR**0000679 (1:112e)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
30C85,
31A15

Retrieve articles in all journals with MSC: 30C85, 31A15

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1985-0768733-1

Article copyright:
© Copyright 1985
American Mathematical Society