Access the full text.
Sign up today, get DeepDyve free for 14 days.
Loading next page...
/lp/springer-journals/level-curve-configurations-and-conformal-equivalence-of-meromorphic-QDOJqP8MqS
References (20)
-
Kenneth Stephenson (1986)
Analytic Functions Sharing Level Curves and TractsAnnals of Mathematics, 123
-
A. Kirillov (1998)
Geometric approach to discrete series of unirreps for Vir.Journal de Mathématiques Pures et Appliquées, 77
-
G. Piranian (1966)
The shape of level curves, 17
-
J. Lint (1997)
Functions of one complex variable II, 40
-
Beardon, Carne, Ng (2002)
The Critical Values of a PolynomialConstructive Approximation, 18
-
M. Cartwright (1938)
On the Level Curves of Integral and Meromorphic FunctionsProceedings of The London Mathematical Society
-
E. Ihrig, M. Ismail (1981)
A -umbral calculus -
Alfredo Poirier (1993)
On postcritically finite polynomials, part 2: Hubbard treesarXiv: Dynamical Systems
-
(1987)
Kähler structure on the k-orbits of a group of diffeomorphisms of the circle -
Kenneth Stephenson, C. Sundberg (1985)
Level Curves of Inner FunctionsProceedings of The London Mathematical Society
-
M Heins (1981)
Meromorphic functions on $$\mathbb{C}$$ C whose moduli have a level set in commonHokkaido Math. J., 10
-
(1937)
Sur les courbes de module constant des fonctions entieres -
J. Walsh (1933)
Note on the location on the critical points of Green's functionBulletin of the American Mathematical Society, 39
-
P. Ebenfelt, D. Khavinson, H. Shapiro (2010)
Two-dimensional shapes and lemniscatesarXiv: Complex Variables
-
W. Hayman, W. Hayman, J. Wu, J. Wu (1981)
Level sets of univalent functionsCommentarii Mathematici Helvetici, 56
-
A. Goodman (1966)
On the convexity of the level curves of a polynomial, 17
-
JL Walsh (1950)
The Location of Critical Points -
P Erdös, F Herzog, G Piranian (1958)
Metric properties of polynomialsJ. Analyse Math., 6
-
(1981)
Meromorphic functions on C whose moduli have a level set in common -
P. Erdös, F. Herzog, G. Piranian (1958)
Metric properties of polynomialsJournal d’Analyse Mathématique, 6
- Publisher
- Springer Journals
- Copyright
- Copyright © 2015 by Springer-Verlag Berlin Heidelberg
- Subject
- Mathematics; Analysis; Computational Mathematics and Numerical Analysis; Functions of a Complex Variable
- ISSN
- 1617-9447
- eISSN
- 2195-3724
- DOI
- 10.1007/s40315-015-0111-5
- Publisher site
- See Article on Publisher Site
Abstract
Let $$f=B_1/B_2$$ f = B 1 / B 2 be a ratio of finite Blaschke products having no critical points on $$\partial \mathbb {D}$$ ∂ D . Then $$f$$ f has finitely many critical level curves (level curves containing critical points of $$f$$ f ) in the disk, and the non-critical level curves interpolate smoothly between the critical level curves. Thus, to understand the geometry of all the level curves of $$f$$ f , one needs only understand the configuration of the finitely many critical level curves of $$f$$ f . In this paper, we show that in fact such a function $$f$$ f is determined not just geometrically but conformally by the configuration of its critical level curves. That is, if $$f_1$$ f 1 and $$f_2$$ f 2 have the same configuration of critical level curves, then there is a conformal map $$\phi $$ ϕ such that $$f_1\equiv f_2\circ \phi $$ f 1 ≡ f 2 ∘ ϕ . We then use this to show that every configuration of critical level curves which could come from an analytic function is instantiated by a polynomial. We also include a new proof of a theorem of Bôcher (which is an extension of the Gauss–Lucas theorem to rational functions) using level curves.
Journal
Computational Methods and Function Theory – Springer Journals
Published: Mar 31, 2015