Harmonic Mappings in the Plane

Peter Duren

doi:10.1017/CBO9780511546600

References

Abu-Muhanna, Y. and Lyzzaik, A., A geometric criterion for decomposition and multivalence, Math. Proc. Cambridge Philos. Soc. 103 (1988), 487–495

Abu-Muhanna, Y. and Lyzzaik, A., The boundary behaviour of harmonic univalent maps, Pacific J. Math. 141 (1990), 1–20

Abu-Muhanna, Y. and Schober, G., Harmonic mappings onto convex domains, Canad. J. Math. 39 (1987), 1489–1530

L. V. Ahlfors, Lectures on Quasiconformal Mappings (Van Nostrand, Princeton, N.J., 1966)

L. V. Ahlfors, Conformal Invariants: Topics in Geometric Function Theory (McGraw-Hill, New York, 1973)

L. V. Ahlfors, Complex Analysis (Third Edition, McGraw-Hill, New York, 1979)

Avci, Y. and Złotkiewicz, E., On harmonic univalent functions, Ann. Univ. Mariae Curie-Skłodowska 44 (1990), 1–7

J. L. M. Barbosa and A. G. Colares, Minimal Surfaces in R3, Lecture Notes in Math. No. 1195 (Springer-Verlag, Berlin, 1986)

Bers, L., Isolated singularities of minimal surfaces, Ann. of Math. 53 (1951), 364–386

Bers, L., Univalent solutions of linear elliptic systems, Comm. Pure Appl. Math. 6 (1953), 513–526

Bojarski, B. V., Homeomorphic solutions of a Beltrami system, Dokl. Akad. Nauk SSSR 102 (1955), 661–664 (Russian)

Bojarski, B. V., On solutions of a linear elliptic system of differential equations in the plane, Dokl. Akad. Nauk SSSR 102 (1955), 871–874 (Russian)

Bojarski, B. V. and Iwaniec, T., Quasiconformal mappings and non-linear elliptic equations in two variables I, II, Bull. Polish Acad. Sci. Math. 22 (1974), 473–478, 479–484

Bshouty, D., Hengartner, N., and Hengartner, W., A constructive method for starlike harmonic mappings, Numer. Math. 54 (1988), 167–178

D. Bshouty and W. Hengartner, Boundary correspondence of univalent harmonic mappings from the unit disc onto a Jordan domain, in Approximation by Solutions of Partial Differential Equations, Quadrature Formulas, and Related Topics, B. Fuglede et al., eds., NATO ASI Series C, Vol. 365 (Kluwer Academic Publishers, Dordrecht-Boston-London, 1992), pp. 51–60

Bshouty, D. and Hengartner, W., Univalent solutions of the Dirichlet problem for ring domains, Complex Variables Theory Appl. 21 (1993), 159–169

Bshouty, D. and Hengartner, W., Univalent harmonic mappings in the plane, Ann. Univ. Mariae Curie-Skłodowska 48 (1994), 12–42

Bshouty, D. and Hengartner, W., Boundary values versus dilatations of harmonic mappings, J. Analyse Math. 72 (1997), 141–164

Bshouty, D. and Hengartner, W., Exterior univalent harmonic mappings with finite Blaschke dilatations, Canad. J. Math. 51 (1999), 470–487

Bshouty, D., Hengartner, W., and Hossian, O., Harmonic typically real mappings, Math. Proc. Cambridge Philos. Soc. 119 (1996), 673–680

Bshouty, D., Hengartner, W., and Suez, T., The exact bound on the number of zeros of harmonic polynomials, J. Analyse Math. 67 (1995), 207–218

Chen, H., Gauthier, P. M., and Hengartner, W., Bloch constants for planar harmonic mappings, Proc. Amer. Math. Soc. 128 (2000), 3231–3240

Choquet, G., Sur un type de transformation analytique généralisant la représentation conforme et définie au moyen de fonctions harmoniques, Bull. Sci. Math. 69 (1945), 156–165

Choquet, G., Sur les homéomorphies harmoniques d'un disque D sur D, Complex Variables Theory Appl. 24 (1993), 47–48

Chuaqui, M., Duren, P., and Osgood, B., The Schwarzian derivative for harmonic mappings, J. Analyse Math. 91 (2003), 329–351

Cima, J. A. and Livingston, A. E., Integral smoothness properties of some harmonic mappings, Complex Variables Theory Appl. 11 (1989), 95–110

Cima, J. A. and Livingston, A. E., Nonbasic harmonic maps onto convex wedges, Colloq. Math. 66 (1993), 9–22

Clunie, J. and Sheil-Small, T., Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A.I 9 (1984), 3–25

Colonna, F., The Bloch constant of bounded harmonic mappings, Indiana Univ. Math. J. 38 (1989), 829–840

Vries, H. L., A remark concerning a lemma of Heinz on harmonic mappings, J. Math. Mech. 11 (1962), 469–471

Vries, H. L., Über Koeffizientenprobleme bei Eilinien und über die Heinzsche Konstante, Math. Z. 112 (1969), 101–106

U. Dierkes, S. Hildebrandt, A. Küster, and O. Wohlrab, Minimal Surfaces I (Springer-Verlag, Berlin-Heidelberg-New York, 1991)

Dorff, M. J., Some harmonic n-slit mappings, Proc. Amer. Math. Soc. 126 (1998), 569–576

Dorff, M. J., Convolutions of planar harmonic convex mappings, Complex Variables Theory Appl. 45 (2001), 263–271

Dorff, M. J., Minimal graphs in ℝ3 over convex domains, Proc. Amer. Math. Soc. 132 (2004), 491–498

Dorff, M. J. and Suffridge, T. J., The inner mapping radius of harmonic mappings of the unit disk, Complex Variables Theory Appl. 33 (1997), 97–103

Dorff, M. and Szynal, J., Harmonic shears of elliptic integrals, Rocky Mountain J. Math., to appear

Driver, K. and Duren, P., Harmonic shears of regular polygons by hypergeometric functions, J. Math. Anal. Appl. 239 (1999), 72–84

P. L. Duren, Theory of HpSpaces (Academic Press, New York, 1970; reprinted with supplement by Dover Publications, Mineola, N.Y., 2000)

P. L. Duren, Univalent Functions (Springer-Verlag, New York, 1983)

P. L. Duren, A survey of harmonic mappings in the plane, in Texas Tech University, Mathematics Series, Visiting Scholars' Lectures 1990–1992, vol. 18 (1992), pp. 1–15

Duren, P. and Hengartner, W., A decomposition theorem for planar harmonic mappings, Proc. Amer. Math. Soc. 124 (1996), 1191–1195

Duren, P. and Hengartner, W., Harmonic mappings of multiply connected domains, Pacific J. Math. 180 (1997), 201–220

Duren, P., Hengartner, W., and Laugesen, R. S., The argument principle for harmonic functions, Amer. Math. Monthly 103 (1996), 411–415

Duren, P. and Khavinson, D., Boundary correspondence and dilatation of harmonic mappings, Complex Variables Theory Appl. 33 (1997), 105–111

Duren, P. and Schober, G., A variational method for harmonic mappings onto convex regions, Complex Variables Theory Appl. 9 (1987), 153–168

Duren, P. and Schober, G., Linear extremal problems for harmonic mappings of the disk, Proc. Amer. Math. Soc. 106 (1989), 967–973

Duren, P. and Thygerson, W. R., Harmonic mappings related to Scherk's saddle-tower minimal surfaces, Rocky Mountain J. Math. 30 (2000), 555–564

Finn, R. and Osserman, R., On the Gauss curvature of non-parametric minimal surfaces, J. Analyse Math. 12 (1964), 351–364

FitzGerald, C. and Pommerenke, Ch., The de Branges theorem on univalent functions, Trans. Amer. Math. Soc. 290 (1985), 683–690

W. H. J. Fuchs, Topics in the Theory of Functions of One Complex Variable (Van Nostrand, Princeton, N.J., 1967)

Gleason, S. and Wolff, T. H., Lewy's harmonic gradient maps in higher dimensions, Comm. Partial Differential Equations 16 (1991), 1925–1968

G. M. Goluzin, Geometric Theory of Functions of a Complex Variable (Moscow, 1952; German transl., Deutscher Verlag, Berlin, 1957; Second Edition, Izdat. “Nauka”, Moscow, 1966; English transl., American Mathematical Society, Providence, R.I., 1969)

Goodloe, M. R., Hadamard products of convex harmonic mappings, Complex Variables Theory Appl. 47 (2002), 81–92

Goodman, A. W. and Saff, E. B., On univalent functions convex in one direction, Proc. Amer. Math. Soc. 73 (1979), 183–187

P. Greiner, Boundary properties of planar harmonic mappings, Ph. D. Thesis, University of Michigan, 1995

Greiner, P., Geometric properties of harmonic shears, Computational Methods and Function Theory, to appear

Grigoryan, A. and Nowak, M., Integral means of harmonic mappings, Ann. Univ. Mariae Curie-Skł, odowska 52 (1998), 25–34

Grigoryan, A. and Nowak, M., Estimates of integral means of harmonic mappings, Complex Variables Theory Appl. 42 (2000), 151–161

Grigoryan, A. and Szapiel, W., Two-slit harmonic mappings, Ann. Univ. Mariae Curie-Skł, odowska 49 (1995), 59–84

Hall, R. R., On a conjecture of Shapiro about trigonometric series, J. London Math. Soc. 25 (1982), 407–415

Hall, R. R., On an inequality of E. Heinz, J. Analyse Math. 42 (1982/83), 185–198

Hall, R. R., A class of isoperimetric inequalities, J. Analyse Math. 45 (1985), 169–180

Hall, R. R., The Gaussian curvature of minimal surfaces and Heinz' constant, J. Reine Angew. Math. 502 (1998), 19–28

W. K. Hayman, Multivalent Functions (Cambridge University Press, London, 1958)

Heinz, E., Über die Lösungen der Minimalflächengleichung, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. (1952), 51–56

Heinz, E., On one-to-one harmonic mappings, Pacific J. Math. 9 (1959), 101–105

Hengartner, W. and Schober, G., On schlicht mappings to domains convex in one direction, Comm. Math. Helv. 45 (1970), 303–314

Hengartner, W. and Schober, G., A remark on level curves for domains convex in one direction, Appl. Analysis 3 (1973), 101–106

Hengartner, W. and Schober, G., Univalent harmonic mappings onto parallel slit domains, Michigan Math. J. 32 (1985), 131–134

Hengartner, W. and Schober, G., On the boundary behavior of orientation-preserving harmonic mappings, Complex Variables Theory Appl. 5 (1986), 197–208

Hengartner, W. and Schober, G., Harmonic mappings with given dilatation, J. London Math. Soc. 33 (1986), 473–483

Hengartner, W. and Schober, G., Univalent harmonic functions, Trans. Amer. Math. Soc. 299 (1987), 1–31

W. Hengartner and G. Schober, Curvature estimates for some minimal surfaces, in Complex Analysis: Articles Dedicated to Albert Pfluger on the Occasion of his 80th Birthday, J. Hersch and A. Huber, editors (Birkhäuser Verlag, Basel, 1988), pp. 87–100

Hengartner, W. and Schober, G., Univalent harmonic exterior and ring mappings, J. Math. Anal. Appl. 156 (1991), 154–171

Hengartner, W. and Szynal, J., Univalent harmonic ring mappings vanishing on the interior boundary, Canad. J. Math. 44 (1992), 308–323

Hersch, J. and Pfluger, A., Généralisation du lemme de Schwarz et du principe de la mesure harmonique pour les fonctions pseudo-analytiques, C. R. Acad. Sci. Paris 234 (1952), 43–45

Hildebrandt, S. and Sauvigny, F., Embeddedness and uniqueness of minimal surfaces solving a partially free boundary value problem, J. Reine Angew. Math. 422 (1991), 69–89

Hoffman, D., The computer-aided discovery of new embedded minimal surfaces, Math. Intelligencer 9 (1987), 8–21

Hopf, E., On an inequality for minimal surfaces z = z(x, y), J. Rational Mech. Anal. 2 (1953), 519–522; 801–802

Jahangiri, J. M., Morgan, C., and Suffridge, T. J., Construction of close-to-convex harmonic polynomials, Complex Variables Theory Appl. 45 (2001), 319–326

S. H. Jun, Harmonic mappings and applications to minimal surfaces, Ph. D. Thesis, Indiana University, 1989

Jun, S. H., Curvature estimates for minimal surfaces, Proc. Amer. Math. Soc. 114 (1992), 527–533

Jun, S. H., Univalent harmonic mappings on Δ = }z: |z| > 1{, Proc. Amer. Math. Soc. 119 (1993), 109–114

Jun, S. H., Planar harmonic mappings and curvature estimates, J. Korean Math. Soc. 32 (1995), 803–814

Khavinson, D. and Swiatek, G., On the number of zeros of certain harmonic polynomials, Proc. Amer. Math. Soc. 131 (2003), 409–414

Kneser, H., Lösung der Aufgabe 41, Jahresber. Deutsch. Math.-Verein. 35 (1926), 123–124

Krzyż, J., and Nowak, M., Harmonic automorphisms of the unit disk, J. Comput. Appl. Math. 105 (1999), 337–346

Laugesen, R. S., Injectivity can fail for higher-dimensional harmonic extensions, Complex Variables Theory Appl. 28 (1996), 357–369

Laugesen, R. S., Planar harmonic maps with inner and Blaschke dilatations, J. London Math. Soc. 56 (1997), 37–48

O. Lehto, Univalent Functions and Teichmüller Spaces (Springer-Verlag, New York, 1987)

O. Lehto and K. I. Virtanen, Quasiconformal Mappings in the Plane (Second Edition, Springer-Verlag, Berlin-Heidelberg-New York, 1973)

Lewy, H., On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc. 42 (1936), 689–692

Lewy, H., On the non-vanishing of the jacobian of a homeomorphism by harmonic gradients, Ann. of Math. 88 (1968), 518–529

Liu, H. and Liao, G., A note on harmonic maps, Appl. Math. Lett. 9 (1996), no. 4, 95–97

Livingston, A. E., Univalent harmonic mappings, Ann. Polon. Math. 57 (1992), 57–70

Livingston, A. E., Univalent harmonic mappings II, Ann. Polon. Math. 67 (1997), 131–145

Lyzzaik, A., On the valence of some classes of harmonic maps, Math. Proc. Cambridge Philos. Soc. 110 (1991), 313–325

Lyzzaik, A., Local properties of light harmonic mappings, Canad. J. Math. 44 (1992), 135–153

Lyzzaik, A., The geometry of some classes of folding polynomials, Complex Variables Theory Appl. 20 (1992), 145–155

Lyzzaik, A., Univalence criteria for harmonic mappings in multiply-connected domains, J. London Math. Soc. 58 (1998), 163–171

Lyzzaik, A., A note on the valency of harmonic maps, J. Math. Anal. Appl. 218 (1998), 611–620

Lyzzaik, A., The modulus of the image annuli under univalent harmonic mappings and a conjecture of J. C. C. Nitsche, J. London Math. Soc. 64 (2001), 369–384

Martio, O., On harmonic quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A.I, (1968), 3–10

Mateljević, M. and Pavlović, M., Multipliers of Hp and BMOA, Pacific J. Math. 146 (1990), 71–84

Melas, A. D., An example of a harmonic map between Euclidean balls, Proc. Amer. Math. Soc. 117 (1993), 857–859

Nehari, Z., The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545–551

Z. Nehari Conformal Mapping (McGraw-Hill, New York, 1952; reprinted by Dover Publications, New York, 1975)

Neumann, G., Valence of complex-valued planar harmonic functions, to appear

Nitsche, J. C. C., Über eine mit der Minimalflächengleichung zusammenhängende analytische Funktion und den Bernsteinschen Satz, Archiv der Math. (Basel) 7 (1956), 417–419

Nitsche, J. C. C., On harmonic mappings, Proc. Amer. Math. Soc. 9 (1958), 268–271

Nitsche, J. C. C., On an estimate for the curvature of minimal surfaces z = z(x, y), J. Math. Mech. 7 (1958), 767–769

Nitsche, J. C. C., On the constant of E. Heinz, Rend. Circ. Mat. Palermo 8 (1959), 178–181

Nitsche, J. C. C., On the module of doubly-connected regions under harmonic mappings, Amer. Math. Monthly 69 (1962), 781–782

Nitsche, J. C. C., Zum Heinzschen Lemma über harmonische Abbildungen, Arch. Math. (Basel) 14 (1963), 407–410

J. C. C. Nitsche, Lectures on Minimal Surfaces, Vol. I (Cambridge University Press, Cambridge, 1989)

Nowak, M., Integral means of univalent harmonic maps, Ann. Univ. Mariae Curie-Skł, odowska 50 (1996), 155–162

Osserman, R., On the Gauss curvature of minimal surfaces, Trans. Amer. Math. Soc. 96 (1960), 115–128

Osserman, R., Global properties of classical minimal surfaces, Duke Math. J. 32 (1965), 565–573

R. Osserman, A Survey of Minimal Surfaces (Second Edition, Dover Publications, Mineola, N.Y., 1986)

R. Osserman, Minimal surfaces in R3, in Global Differential Geometry, S. S. Chern, editor, Mathematical Association of America Studies in Mathematics Vol. 27 (1989), pp. 73–98

G. Pólya and G. Szegő, Isoperimetric Inequalities in Mathematical Physics (Princeton University Press, Princeton, N.J., 1951)

Ch. Pommerenke, Univalent Functions (Vandenhoeck & Ruprecht, Göttingen, 1975)

Radó, T., Aufgabe 41, Jahresber. Deutsch. Math.-Verein. 35 (1926), 49

Radó, T., Über den analytischen Charakter der Minimalflächen, Math. Z. 24 (1926), 321–327

Radó, T., Zu einem Satze von S. Bernstein über Minimalflächen im Grossen, Math. Z. 26 (1927), 559–565

Reich, E., The composition of harmonic mappings, Ann. Acad. Sci. Fenn. Ser. A.I 12 (1987), 47–53

Reich, E., Local decomposition of harmonic mappings, Complex Variables Theory Appl. 9 (1987), 263–269

H. Renelt, Quasikonforme Abbildungen und elliptische Systeme erster Ordnung in der Ebene (B. G. Teubner, Leipzig, 1982); English edition: Elliptic Systems and Quasiconformal Mappings (John Wiley & Sons, New York, 1988)

Robertson, M. S., Analytic functions star-like in one direction, Amer. J. Math. 58 (1936), 465–472

Royster, W. C. and Ziegler, M., Univalent functions convex in one direction, Publ. Math. Debrecen 23 (1976), 339–345

W. Rudin, Principles of Mathematical Analysis (Third Edition, McGraw-Hill, New York, 1976)

Ruscheweyh, St. and Salinas, L., On the preservation of direction-convexity and the Goodman-Saff conjecture, Ann. Acad. Sci. Fenn. Ser. A.I 14 (1989), 63–73

Schaubroeck, L. E., Subordination of planar harmonic functions, Complex Variables Theory Appl. 41 (2000), 163–178

Schaubroeck, L. E., Growth, distortion and coefficient bounds for plane harmonic mappings convex in one direction, Rocky Mountain J. Math. 31 (2001), 625–639

G. Schober, Planar harmonic mappings, in Computational Methods and Function Theory, Lecture Notes in Math. No. 1435 (Springer-Verlag, Berlin-Heidelberg, 1990), pp. 171–176

J. T. Schwartz, Nonlinear Functional Analysis (Gordon and Breach, New York, 1969)

H. S. Shapiro, Research problems in complex analysis (edited by J. M. Anderson, K. F. Barth, and D. A. Brannan), Bull. London Math. Soc. 9 (1977), 129–162; Problem No. 7.26, p. 146

Sheil-Small, T., On the Fourier series of a finitely described convex curve and a conjecture of H. S. Shapiro, Math. Proc. Cambridge Philos. Soc. 98 (1985), 513–527

Sheil, T.-Small, On the Fourier series of a step function, Michigan Math. J. 36 (1989), 459–475

Sheil, T.-Small, Constants for planar harmonic mappings, J. London Math. Soc. 42 (1990), 237–248

J. J. Stoker, Differential Geometry (Wiley-Interscience, New York, 1969)

D. J. Struik, Lectures on Classical Differential Geometry (Second Edition, Addison-Wesley, Cambridge, Mass., 1961; reprinted by Dover Publications, Mineola, N.Y., 1988)

Suffridge, T. J., Harmonic univalent polynomials, Complex Variables Theory Appl. 35 (1998), 93–107

Suffridge, T. J. and Thompson, J. W., Local behavior of harmonic mappings, Complex Variables Theory Appl. 41 (2000), 63–80

Szulkin, A., An example concerning the topological character of the zero-set of a harmonic function, Math. Scand. 43 (1978), 60–62

Ullman, J. L. and Titus, C. J., An integral inequality with applications to harmonic mappings, Michigan Math. J. 10 (1963), 181–192

N. Vekua, Generalized Analytic Functions (Pergamon Press, London, 1962)

A. Weitsman, Harmonic mappings whose dilatations are singular inner functions, preprint (1996)

Weitsman, A., On the dilatation of univalent planar harmonic mappings, Proc. Amer. Math. Soc. 126 (1998), 447–452

Weitsman, A., On the Fourier coefficients of homeomorphisms of the circle, Math. Res. Lett. 5 (1998), 383–390

Weitsman, A., On univalent harmonic mappings and minimal surfaces, Pacific J. Math. 192 (2000), 191–200

Weitsman, A., Univalent harmonic mappings of annuli and a conjecture of J. C. C. Nitsche, Israel J. Math. 124 (2001), 327–331

Weitsman, A., On the Poisson integral of step functions and minimal surfaces, Canad. Math. Bull. 45 (2002), 154–160

W. L. Wendland, Elliptic Systems in the Plane (Pitman, London, 1979)

Wilmshurst, A., The valence of harmonic polynomials, Proc. Amer. Math. Soc. 126 (1998), 2077–2081

Wood, J. C., Lewy's theorem fails in higher dimensions, Math. Scand. 69 (1991), 166

Harmonic Mappings in the Plane

This Book has been cited by the following publications. This list is generated based on data provided by Crossref.

Book description

Reviews

Refine List

Actions for selected content:

Contents

Frontmatter
pp i-vi

Contents
pp vii-x

Preface
pp xi-xii

1 - Preliminaries
pp 1-17

2 - General Properties of Harmonic Mappings
pp 18-28

3 - Harmonic Mappings onto Convex Regions
pp 29-56

4 - Harmonic Self-Mappings of the Disk
pp 57-77

5 - Harmonic Univalent Functions
pp 78-88

6 - Extremal Problems
pp 89-110

7 - Mapping Problems
pp 111-135

8 - Additional Topics
pp 136-155

9 - Minimal Surfaces
pp 156-172

10 - Curvature of Minimal Surfaces
pp 173-195

Appendix: Extremal Length
pp 196-200

References
pp 201-210

Index
pp 211-212

Metrics

Altmetric attention score

Full text views

Book summary page views

Accessibility standard: Unknown

Why this information is here

Accessibility Information

Harmonic Mappings in the Plane

Book description

Reviews

Refine List

Actions for selected content:

Save Search

Contents

Metrics

Altmetric attention score

Full text views

Book summary page views

Accessibility standard: Unknown

Why this information is here

Accessibility Information