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9780444515476

Handbook of Complex Analysis

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  • ISBN13:

    9780444515476

  • ISBN10:

    044451547X

  • Format: Hardcover
  • Copyright: 2004-12-30
  • Publisher: Elsevier Science
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Summary

Geometric Function Theory is that part of Complex Analysis which covers the theory of conformal and quasiconformal mappings. Beginning with the classical Riemann mapping theorem, there is a lot of existence theorems for canonical conformal mappings. On the other side there is an extensive theory of qualitative properties of conformal and quasiconformal mappings, concerning mainly a prior estimates, so called distortion theorems (including the Bieberbach conjecture with the proof of the Branges). Here a starting point was the classical Scharz lemma, and then Koebe's distortion theorem. There are several connections to mathematical physics, because of the relations to potential theory (in the plane). The Handbook of Geometric Function Theory contains also an article about constructive methods and further a Bibliography including applications eg: to electroxtatic problems, heat conduction, potential flows (in the plane).

Table of Contents

Preface v
List of Contributors of Volume 1 vii
List of Contributors ix
Contents of Volume 1 xi
1. Quasiconformal mappings in Euclidean spaces
1(30)
F.W. Gehring
1. Definitions
3(1)
2. Historical remarks
4(1)
2.1. Mappings in the plane
4(1)
2.2. Mappings in higher dimensions
4(1)
2.3. Mappings in arbitrary metric spaces
4(1)
3. Role played by quasiconformal mappings
5(1)
4. Tools to study quasiconformal mappings
5(3)
5. Mapping problems
8(2)
6. Extensions of mappings
10(1)
7. Boundary correspondence and lifting
11(1)
8. Measurable Riemann mapping theorem
12(1)
9. Distortion and equicontinuity
13(1)
10. Properties of the Jacobian
14(1)
11. Connections with functional analysis
15(2)
12. Connections with geometry and elasticity
17(1)
13. Connections with complex analysis
18(2)
14. Connections with differential equations
20(1)
15. Connections with topology
21(1)
16. Connections with discrete groups
21(2)
17. An application to medicine
23(1)
References
23(8)
2. Variational principles in the theory of quasiconformal maps
31(68)
S.L. Krushkal
1. Background: Quasiconformal maps and variations
33(66)
1.1. What does quasiconformality mean?
33(1)
1.2. Representation and variation formulas
34(2)
1.3. Other explicit variational formulas
36(2)
1.4. A boundary quasiconformal variation
38(2)
1.5. An old problem of I.N. Vekua
40(1)
2. General theory of extremal quasiconformal maps
41(1)
2.1. Background: The Grötzsch problem
41(1)
2.2. Teichmüller's theory of extremal quasiconformal maps
42(1)
2.3. Geometric picture
44(1)
2.4. The deformation (Teichmüller) space
46(1)
2.5. Topics in complex metric geometry of Teichmüller spaces
47(1)
2.6. General variational problems for quasiconformal maps of Riemann surfaces of finite type
52(1)
2.7. Back to tori and annuli
56(1)
2.8. Extremal quasiconformal maps: General theory
61(1)
2.9. A new general variational principle
65(6)
2.10. Examples
71(2)
2.11. Extremal quasiconformal embeddings
73(5)
2.12. Quasiconformality in the mean
78(2)
3. Nonlinear quasiconformal maps
80(1)
3.1. Lavrentiev-Lindelöf variational principle for strongly elliptic systems
80(1)
3.2. Main theorem for strips
81(1)
4. Quasilinear Beltrami equation
82(1)
4.1. Gutlyanskii-Ryazanov's method
82(3)
5. A glimpse at further methods and developments
85(3)
References
88(11)
3. The conformal module of quadrilaterals and of rings
99(32)
R. Kühnau
1. Definition of the module
101(1)
2. A boundary value problem: Dirichlet's principle
102(1)
3. Capacitance
102(1)
4. Examples, symmetries
103(1)
5. Numerical calculation of the modulus
104(1)
6. Grötzsch's strip method
104(1)
7. Grötzsch's principle
104(1)
8. Simple estimates for the modules
105(1)
9. Some estimates of the module of rings with geometric quantities
105(1)
10. Extremal decomposition problems
106(1)
11. Method of extremal length
107(2)
12. Module of one-parameter curve families
109(1)
13. Small changes of a quadrilateral
110(1)
14. Long quadrilaterals
111(1)
15. Module of a thin worm
111(2)
16. Module and hyperbolic/elliptic transfinite diameter
113(1)
17. Higher dimensions
114(1)
18. Limit cases: Reduced modules
114(1)
19. Symmetrization and other geometric transformations
115(1)
20. Examples of ring domains and quadrilaterals
115(2)
21. Harmonic measure and conformal module
117(3)
22. Conformal module and quasiconformal mappings
120(1)
23. Harmonic mappings
121(1)
24. Inner and outer domain of a Jordan curve
122(2)
25. Miscellaneous. Problems
124(1)
References
125(6)
4. Canonical conformal and quasiconformal mappings. Identities. Kernel functions
131(34)
R. Kühnau
1. Introduction
133(1)
2. Some simple canonical conformal mappings
134(6)
3. Nonuniqueness: "Verzweigungserscheinung" in the sense of Grotzsch
140(1)
4. Koebe's Kreisnormierungs theorem. Circle packings
141(2)
5. Identities between the canonical conformal mappings
143(1)
6. Connections with other fundamental solutions: Green's function, Neumann's function, harmonic measure. Orthonormal series, kernel function
144(1)
7. Conformal mapping of domains Ginfinity of infinite connectivity
145(1)
8. Kernel convergence. Dependence on parameters
146(1)
9. Boundary behavior of the mappings
146(1)
10. Integral equation methods
147(1)
11. Goluzin's functional equation
147(1)
12. Iteration procedures
147(1)
13. Factorization
148(1)
14. Canonical conformal mappings with symmetries: Mappings on the elliptic and on the hyperbolic plane
149(3)
15. Canonical conformal mappings on a fixed Riemann surface
152(1)
16. Canonical conformal mappings with higher normalization
152(2)
17. Numerical realization of canonical conformal mappings
154(1)
18. Generalizations for quasiconformal mappings
154(3)
19. A desideratum: Another way from conformal to quasiconformal mappings
157(1)
20. Miscellaneous
157(2)
References
159(6)
5. Univalent holomorphic functions with quasiconformal extensions (variational approach)
165(78)
S.L. Krushkal
0. Introduction
167(76)
0.1. Interaction between univalent functions and Teichmüller space theory
167(1)
0.2. General remarks on analytic functionals
167(1)
0.3. Remarks on variational methods
168(1)
0.4. New phenomena
168(1)
0.5. Grunsky coefficients
169(1)
0.6. Related quadratic differentials
169(1)
1. The existence theorems for special quasiconformal deformations: Old and new
170(1)
1.1. Two local theorems
170(1)
1.2. Sketch of the proof of Theorem 1.1
171(2)
1.3. Quasiconformal deformations decreasing Lp-norm
173(3)
1.4. Finite boundary interpolation by univalent functions
176(1)
2. Grunsky coefficient inequalities, Carathéodory metric, Fredholm eigenvalues and asymptotically conformal curves
176(1)
2.1. Main theorem
176(3)
2.2. Geometric features
179(1)
2.3. Equivalence of conditions (2.5) and (2.9) for asymptotically conformal curves
180(2)
2.4. Two examples
182(2)
2.5. The Teichmüller-Kühnau extension of univalent functions
184(3)
2.6. The Fredholm eigenvalues
187(1)
3. Distortion theory for univalent functions with quasiconformal extension
187(1)
3.1. General distortion problems for univalent functions with quasiconformal extension
187(1)
3.2. Lehto's majoration principle and its improvements. General range value theorems
188(3)
3.3. Generalization: The maps with dilatations bounded by a nonconstant function
191(1)
3.4. Examples
192(2)
4. General distortion theorems for univalent functions with quasiconformal extension
194(1)
4.1. General variational problem
194(2)
4.2. Generalizations of Theorem 4.1
196(1)
4.3. Lower bound for k0 (F)
196(1)
4.4. Two more illustrative examples
197(1)
5. The coefficient problem for univalent functions with quasiconformal extensions. Small dilatations
198(1)
5.1. Main theorems
198(1)
5.2. Proof of Theorem 5.2
199(4)
5.3. Complementary remarks and open questions
203(1)
6. Other variational methods
204(1)
6.1. A general method of quasiconformal variations
204(2)
6.2. Schiffer's method
206(1)
6.3. Some applications: The Schiffer-Schober and McLeavey distortion theorems
207(2)
6.4. Variations of Kühnau
209(2)
6.5. Variations of Gutlyansky
211(4)
6.6. Applications of the Dirichlet principle and of Fredholm eigenvalues. Kühnau's method. Applications
215(3)
6.7. The Dirichlet principle and the area method
218(3)
6.8. Other methods and results
221(2)
6.9. Multivalent functions
223(1)
7. Univalent functions and universal Teichmüller space
223(1)
7.1. The Bers embedding of universal Teichmüller space
223(2)
7.2. Holomorphic curves in the set of Schwarzian derivatives of univalent functions
225(1)
7.3. Some topological properties
226(1)
7.4. Conformally rigid domains and shape of Teichmüller spaces
227(1)
7.5. Remarks on other holomorphic embeddings of universal Teichmüller space
228(1)
References
229(14)
6. Transfinite diameter, Chebyshev constant and capacity
243(66)
S. Kirsch
1. Introduction
245(1)
2. Alternate descriptions of transfinite diameter
247(62)
2.1. Transfinite diameter
247(2)
2.2. Chebyshev constant
249(1)
2.3. Green function and Robin constant
250(1)
2.4. Logarithmic capacity
251(2)
2.5. Extremal length
253(1)
2.6. Conformal mapping radius
253(1)
3. Estimates of transfinite diameter
254(5)
4. Asymptotic distribution of extremal points and applications
259(1)
4.1. Fekete points
259(1)
4.2. Polynomial interpolation
260(1)
4.3. Fejér points
261(1)
4.4. A summation method in numerical linear algebra
262(1)
4.5. Menke points
263(1)
4.6. Leja points
264(1)
5. Analytic capacity and rational approximation
265(1)
5.1. Analytic capacity
265(3)
5.2. Rational approximation
268(1)
6. Generalizations of logarithmic capacity
269(1)
6.1. Weighted capacity
269(7)
6.2. Hyperbolic capacity
276(4)
6.3. Elliptic capacity
280(2)
6.4. Green capacity
282(3)
6.5. Robin capacity
285(3)
6.6. Capacity and conformal maps of multiply-connected domains
288(3)
6.7. Capacity and quasiconformal maps
291(6)
6.8. Capacity in CN
297(5)
References
302(7)
7. Some special classes of conformal mappings
309(30)
T.J. Suffridge
1. Preliminary results
311(1)
2. Starlike and convex domains
312(1)
3. Coefficient inequalities and growth rates
317(1)
4. Radii of starlikeness and convexity for S
318(1)
5. Further properties of starlike functions
321(1)
6. Close to convex functions
325(1)
7. Spirallike functions
328(1)
8. Typically real functions
330(1)
9. Some integral representations and extreme point theory
334(1)
References
337(2)
8. Univalence and zeros of complex polynomials
339(12)
G. Schmieder
Introduction
341(1)
1. General criteria
341(1)
2. External univalent polynomials
343(1)
3. Univalent trinomials
344(1)
References
349(2)
9. Methods for numerical conformal mapping
351(128)
R. Wegmann
1. Introduction
353(1)
2. Auxiliary material
354(15)
2.1. Spaces
354(1)
2.2. Conformal mapping
355(3)
2.3. Corners
358(1)
2.4. Crowding
359(3)
2.5. Function theoretic boundary value problems
362(5)
2.6. The operator R
367(2)
3. Mapping from the region to the disk
369(18)
3.1. Potential theoretic methods
369(8)
3.2. Extremum principles
377(8)
3.3. Osculation methods
385(1)
3.4. Accuracy
386(1)
4. Mapping from the disk to the region
387(29)
4.1. Mapping to nearby regions
388(1)
4.2. Projection
389(12)
4.3. Newton methods
401(7)
4.4. Interpolation
408(7)
4.5. Accuracy
415(1)
5. Mapping from an ellipse to the region
416(2)
6. Waves
418(1)
7. Mapping from a quadrilateral to a rectangle
419(2)
8. Mapping of exterior regions
421(11)
8.1. Mapping from the exterior region to the exterior of the disk
421(3)
8.2. Mapping from the exterior of the disk to the exterior region
424(8)
9. Mapping to Riemann surfaces
432(5)
10. Mapping of a doubly-connected region to an annulus
437(164)
10.1. Potential theoretic methods
437(2)
10.2. Extremum principles
439(1)
11. Mapping from an annulus to a doubly-connected region
440(1)
11.1. Boundary value problems
441(2)
11.2. Projection
443(3)
11.3. The Newton method
446(4)
11.4. Other methods
450(1)
12. Multiply-connected regions
450(2)
12.1. Potential theoretic methods
452(1)
12.2. Osculation methods
453(2)
12.3. Projection
455(3)
12.4. Riemann-Hilbert problems
458(3)
12.5. The Newton method
461(3)
12.6. Other methods
464(3)
References
467(12)
10. Univalent harmonic mappings in the plane 479(28)
D. Bshouty and W. Hengartner
1. Introduction
481(1)
2. Univalent harmonic mappings on a simply connected domain
485(24)
2.1. Motivation
485(1)
2.2. Univalent harmonic mappings defined on the plane
486(1)
2.3. The classes SH and SºH
487(1)
2.4. Univalent harmonic mappings onto convex domains
488(3)
2.5. Mapping problems
491(2)
2.6. Boundary behavior
493(3)
2.7. Univalent log-harmonic mappings
496(1)
2.8. Constructive methods
497(2)
3. Univalent harmonic mappings on multiply connected domains
499(1)
3.1. Univalent harmonic mapping defined on the exterior of the unit disk
499(2)
3.2. Univalent harmonic ring mappings
501(1)
3.3. Extensions of Kneser's theorem
502(1)
3.4. Canonical harmonic-punctured plane mappings
503(1)
References
504(3)
11. Quasiconformal extensions and reflections 507(48)
S.L. Krushkal
0. Preface
509(1)
1. Holomorphic motions and quasiconformal extension of univalent functions. Kühnau's problems
509(48)
1.1. Some sufficient conditions for quasiconformal extension of univalent functions
509(8)
1.2. Holomorphic motions
517(1)
1.3. Which are the exact bounds for quasiconformal extensions of univalent functions?
517(2)
1.4. Sketch of the proof of relation (1.28)
519(2)
1.5. An improvement of Grunsky's and Milin's univalence criteria
521(1)
1.6. r²-property and a dynamical characterization of the disk
522(1)
1.7. Application to coefficient estimates for univalent functions: Zalcman's conjecture
523(1)
1.8. Analytic dependence of conformal invariants on parameters
524(1)
2. Quasireflections across curves
525(1)
2.1. Topological background
525(1)
2.2. Quasiconformal reflections
526(1)
2.3. Fredholm eigenvalues
527(2)
2.4. Results of Kühnau
529(1)
2.5. Some qualitative estimates
530(1)
2.6. Quasiconformal extension and reflections across quasicircles
531(1)
2.7. Sketch of the proof of Theorem 2.5
532(1)
2.8. Asymptotical approach
533(1)
2.9. Reflection coefficients and Fredholm eigenvalues of convex domains
534(5)
3. The best polynomial approximation of holomorphic functions and general quasiconformal mirrors
539(1)
3.1. Pluricomplex Green function and holomorphic extension
539(1)
3.2. Applications to holomorphic functions on the interval and quasireflections over arcs
540(2)
3.3. General quasiconformal mirrors
542(3)
References
545(10)
12. Beltrami equation 555(44)
U. Srebro and E. Yakubov
1. Introduction and notations
557(44)
1.1. The Beltrami equation
557(1)
1.2. Historical remarks
557(1)
1.3. Applications of Beltrami equations
558(1)
1.4. Classification of Beltrami equations
558(1)
1.5. ACL solutions
559(1)
1.6. Ellipticity of the Beltrami equation
559(1)
2. The classical case: ||μ||infinity less than 1 560
2.1. Quasiconformal mappings
560(1)
2.2. Main problems
560(1)
2.3. Integrability
561(1)
2.4. Methods of proof of uniqueness and existence
562(1)
2.5. Uniqueness
562(1)
2.6. Existence
562(3)
2.7. Smoothness of the solutions
565(1)
2.8. Analytic dependence on parameters
565(1)
3. The relaxed classical case:|μ| less than 1a.e and ||μ||infinity = 1 565
3.1. Approximation
565(1)
3.2. Examples
566(1)
3.3. The singular set
567(1)
3.4. Case (i): The singular set E is specified and E D
568(1)
3.5. Case (ii): The singular set E is specified and E D
572(1)
3.6. Case (iii): The singular set E is not specified
573(1)
3.7. BMO functions
578(1)
3.8. Modulus inequality
579(1)
3.9. The subclasses Da, Tu, BJ and RSY
581
3.10. FMO functions
580(1)
3.11. Example of a function Q FMO\BMOloc
581(1)
4. Alternating Beltrami equation
582(1)
4.1. Introduction
582(1)
4.2. The geometric configuration and the conditions on μ
582(1)
4.3. A symmetric form of the alternating Beltrami equation
583(1)
4.4. Branched folded maps
583(1)
4.5. Definition of BF-maps
583(1)
4.6. Discrete maps, canonical maps and classification of critical points
584(1)
4.7. Branch points, power maps and winding maps
584(1)
4.8. Folding maps
584(1)
4.9. (p, q)-cusp maps and (p, q)-cusp points
585(1)
4.10. Umbrella and simple umbrella maps
585(1)
4.11. Alternating Beltrami equations and FOR-maps
586(1)
4.12. Proper folding, cusp and umbrella solutions
586(2)
4.13. Existence of local folding solutions
588(1)
4.14. Uniformization and folds
589(1)
5. Open problems
590(1)
5.1. Problem 1
590(1)
5.2. Problem 2
591(1)
5.3. Problem 3
591(1)
5.4. Problem 4
591(1)
5.5. Problem 5
591(1)
Acknowledgments
591(1)
References
592(7)
13. The application of conformal maps in electrostatics 599(22)
R. Kühnau
1. Introduction
601(2)
2. Results
603(1)
3. The underlying main inequalities
604(1)
4. A lemma for conformal mapping of an annulus
605(1)
5. A lemma for area distortion in the complex plane
606(2)
6. Proof of Theorem 1
608(3)
7. Proof of Theorem 2 (including also the case F* = infinity)
611(1)
8. Proof of Theorem 3 (2 is convex)
611(1)
9. Estimation in the other direction
612(4)
10. Special case: Leyden jar
616(3)
References
619(4)
Introduction
623
14. Special functions in Geometric Function Theory 621(40)
S.-L. Qiu and M. Vuorinen
1. Gamma and beta functions
624(39)
1.1. Functional equalities
624(1)
1.2. Euler-Mascheroni constant γ
625(1)
1.3. The psi function
626(1)
1.4. Functional equalities of ψ(z)
626(1)
1.5. Special values Γ(z) and ψ(z)
627(1)
1.6. The Appell symbol
627(1)
1.7. The Stirling and Wallis formulas
628(1)
1.8. Asymptotic formulas
628(1)
2. Hypergeometric functions
629(1)
2.1. Integral representation
629(1)
2.2. Elementary particular cases
630(1)
2.3. Differentiation formula
630(1)
2.4. Special values of the argument
630(1)
2.5. Hypergeometric differential equation
631(1)
2.6. Gauss' contiguous relations
631(1)
2.7. Transformation formulas
633(1)
2.8. Properties of coefficients of hypergeometric series
634(1)
2.9. Asymptotic and monotonicity properties of F(a, b; c; z)
635(1)
2.10. More expansions
636(1)
2.11. Historical remarks
636(1)
2.12. Classification of functions
637(1)
3. Complete elliptic integrals
637(1)
3.1. Definition of complete elliptic integrals
637(1)
3.2. Arc length of an ellipse
638(1)
3.3. Generalized elliptic integrals
639(1)
3.4. Identity and derivative formulas
639(1)
3.5. Particular values
640(1)
3.6. Landen's identities
640(1)
3.7. Elliptic integral algorithm
640(1)
3.8. The class Σ
641(1)
3.9. Theta functions
641(1)
3.10. Incomplete elliptic integrals
642(1)
3.11. Historical remarks
642(1)
4. Quotients of elliptic integrals
642(1)
4.1. Identities and derivative formulas
643(2)
4.2. Expansions
645(1)
4.3. Generalization of μ(r)
646(1)
4.4. The φ-distortion function
646(1)
4.5. Modular equations
647(1)
4.6. Singular values
647(1)
4.7. Schwarz' lemma for quasiconformal maps
648(1)
5. Elliptic functions
648(1)
5.1. Doubly-periodic functions
649(1)
5.2. Elliptic functions
649(1)
5.3. Special values
650(1)
5.4. Squared relations
650(1)
5.5. Derivatives
650(1)
5.6. Addition formulas
651(1)
5.7. Double and half arguments
651(1)
5.8. Jacobi's imaginary transformation
652(1)
5.9. Complex arguments
652(1)
5.10. Moduli of the elliptic functions
653(1)
5.11. Periodicity properties
653(1)
5.12. Maclaurin's series
654(1)
5.13. Poles, residues and zeros
654(1)
5.14. Landen's transformations
654(1)
5.15. Theta function formulas for elliptic functions
655(1)
5.16. Occurrence in applications
655(1)
5.17. Historical remarks
655(1)
5.18. Computer algorithms for elliptic functions
656(1)
References
656(5)
15. Extremal functions in Geometric Function Theory. Higher transcendental functions. Inequalities 661(8)
R. Kühnau
1. Introduction
663(1)
2. Some examples of special functions in Geometric Function Theory
664(1)
3. A curiosity: Deriving real inequalities from results in Geometric Function Theory
665(1)
References
667(2)
16. Eigenvalue problems and conformal mapping 669(18)
B. Dittmar
1. Introduction
671(1)
2. Isoperimetric inequalities
671(18)
2.1. Membrane problems
671(6)
2.2. Stekloff problem
677(3)
2.3. Mixed Stekloff eigenvalues
680(2)
2.4. Trilaterals, quadrilaterals and their eigenvalues
682(1)
2.5. Numerical calculation
683(1)
References
683(4)
17. Foundations of quasiconformal mappings 687(68)
C. Andreian Cazacu
Introduction
689(1)
1. Differentiability properties
693(64)
1.1 Definitions of differentiable mappings
693(1)
1.2 Dilatation quotient
694(1)
1.3 Complex dilatation and Beltrami equation
695(2)
2. Modules and extremal length
697(1)
2.1 Module of a quadrilateral and of a ring domain
697(2)
2.2 Extremal length and module of curve family
699(1)
2.3 ρ module
699(3)
2.4 Module with weight, generalizations and length-area dilation
702(3)
2.5 Connections and generalizations
705(1)
3. Grötzsch qc and qr mappings
706(1)
3.1 Grötzsch's work
706(3)
3.2 Quasiconformal mappings in Ahfors' value distribution
709(1)
3.3 Lavrent'ev's almost analytic mappings
710(4)
3.4 Teichmüller's results
714(8)
4. General qc and qr mappings
722(1)
4.1 Geometric definition
722(7)
4.2 Analytic definition
729(4)
4.3 Metric definition
733(2)
4.4 Other definition
735(1)
Acknowledgment
736(1)
References
736(19)
18. Quasiconformal mappings in value-distribution theory 755(54)
D. Drasin, A.A. Gol'dberg and P. Poggi-Corradini
1. Introduction
757(72)
1.1. Introduction
757(1)
1.2. Nevanlinna theory
758(1)
2. Background
759(1)
2.1. Early history
759(3)
2.2. Nevanlinna's class Fq
762(2)
2.3. Enter Ahlfors
764(4)
3. The type problem: Introduction
768(1)
3.1. The type problem. Speiser graphs
768(2)
3.2. The Nevanlinna-Witlich criterion
770(1)
3.3. A necessary condition for pataholicity
771(1)
4. The type problem: Basic methods
772(1)
4.1. General methods in the type problem
772(3)
4.2. Interlude: A qc exhaustion and a weak converse of Theorem 3
775(5)
4.3. Some model classes of surfaces
780(4)
4.4. Spiraling
784(5)
4.5. Parking-Garage surfaces
789(2)
5. Nevanlinna theory: Classical methods
791(1)
5.1. On a problem of Nevanlinna
791(1)
5.2. An application to surfaces in R³
792(1)
5.3. The Teichmüller-Wittich-Belinskii theorem
792(2)
6. Nevanlinna theory: Modem developments. Miscellany
794(1)
6.1. Nevanlinna's inverse problem
794(1)
6.2. Inverse problem: Earlier work
795(1)
6.3. The deficiency problem
795(3)
6.4. Lindelöf ends and the solution
798(1)
6.5. F. Nevanlinna conjecture. Extremal functions
799(1)
7. Some recent advances
800(4)
Acknowledgments
804(1)
References
804(5)
19. Bibliography of Geometric Function Theory 809(20)
R. Kühnau
Author Index 829(20)
Subject Index 849

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