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

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

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## 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 $l\left({\phi }_{1}\right)+l\left({\phi }_{2}\right)$ for fixed ${\phi }_{1}$ and ${\phi }_{2}$$\left(0\le {\phi }_{1}\le {\phi }_{2}\le 2\pi \right)$.

## 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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