Radial segments and conformal mapping of an annulus onto domains bounded by a circle and a k-circle

Tetsuo Inoue

Annales Polonici Mathematici (1992)

  • Volume: 56, Issue: 2, page 157-162
  • ISSN: 0066-2216

Abstract

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Let f(z) be a conformal mapping of an annulus A(R) = 1 < |z| < R and let f(A(R)) be a ring domain bounded by a circle and a k-circle. If R(φ) = w : arg w = φ, and l(φ) - 1 is the linear measure of f(A(R)) ∩ R(φ), then we determine the sharp lower bound of l ( φ 1 ) + l ( φ 2 ) for fixed φ 1 and φ 2 ( 0 ≤ φ 1 ≤ φ 2 ≤ 2 π ) .

How to cite

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Tetsuo Inoue. "Radial segments and conformal mapping of an annulus onto domains bounded by a circle and a k-circle." Annales Polonici Mathematici 56.2 (1992): 157-162. <http://eudml.org/doc/262294>.

@article{TetsuoInoue1992,
abstract = {Let f(z) be a conformal mapping of an annulus A(R) = 1 < |z| < R and let f(A(R)) be a ring domain bounded by a circle and a k-circle. If R(φ) = w : arg w = φ, and l(φ) - 1 is the linear measure of f(A(R)) ∩ R(φ), then we determine the sharp lower bound of $l(φ_1) + l(φ_2)$ for fixed $φ_1$ and $φ_2$$(0 ≤ φ_1 ≤ φ_2 ≤ 2π)$.},
author = {Tetsuo Inoue},
journal = {Annales Polonici Mathematici},
keywords = {Koebe region; chordal cross ratio; -circle},
language = {eng},
number = {2},
pages = {157-162},
title = {Radial segments and conformal mapping of an annulus onto domains bounded by a circle and a k-circle},
url = {http://eudml.org/doc/262294},
volume = {56},
year = {1992},
}

TY - JOUR
AU - Tetsuo Inoue
TI - Radial segments and conformal mapping of an annulus onto domains bounded by a circle and a k-circle
JO - Annales Polonici Mathematici
PY - 1992
VL - 56
IS - 2
SP - 157
EP - 162
AB - Let f(z) be a conformal mapping of an annulus A(R) = 1 < |z| < R and let f(A(R)) be a ring domain bounded by a circle and a k-circle. If R(φ) = w : arg w = φ, and l(φ) - 1 is the linear measure of f(A(R)) ∩ R(φ), then we determine the sharp lower bound of $l(φ_1) + l(φ_2)$ for fixed $φ_1$ and $φ_2$$(0 ≤ φ_1 ≤ φ_2 ≤ 2π)$.
LA - eng
KW - Koebe region; chordal cross ratio; -circle
UR - http://eudml.org/doc/262294
ER -

References

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  1. [1] L. V. Ahlfors, Quasiconformal reflections, Acta Math. 109 (1963), 291-301. Zbl0121.06403
  2. [2] D. K. Blevins, Conformal mappings of domains bounded by quasiconformal circles, Duke Math. J. 40 (1973), 877-883. Zbl0275.30015
  3. [3] D. K. Blevins, Harmonic measure and domains bounded by quasiconformal circles, Proc. Amer. Math. Soc. 41 (1973), 559-564. Zbl0281.30015
  4. [4] D. K. Blevins, Covering theorems for univalent functions mapping onto domains bounded by quasiconformal circles, Canad. J. Math. 28 (1976), 627-631. Zbl0362.30013
  5. [5] W. K. Hayman, Multivalent Functions, Cambridge Univ. Press, 1958. 
  6. [6] J. A. Jenkins, Some uniqueness results in the theory of symmetrization, Ann. of Math. 61 (1955), 106-115. Zbl0064.07501
  7. [7] O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane, second ed., Springer, 1973. Zbl0267.30016
  8. [8] I. P. Mityuk, Principle of symmetrization for the annulus and some of its applications, Sibirsk. Mat. Zh. 6 (1965), 1282-1291 (in Russian). 

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