Harmonic Mappings in the Plane

Duren, Peter

doi:10.1017/cbo9780511546600

Cited by 580 publications

(526 citation statements)

References 107 publications

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“…Further information about harmonic mappings and their relation to minimal surfaces can be found in the book [8].…”

Section: Its Most Important Property Is Möbius Invariance: S(t • F )mentioning

confidence: 99%

“…Schwarzian norms of harmonic mappings. Let f = h + g be a locally univalent orientation-preserving harmonic mapping with dilatation g ′ /h ′ = q 2 , where q is analytic and |q(z)| < 1 in the unit disk D. Then the Schwarzian Sf is defined by (7) and has the expression (8). Let Sf denote the hyperbolic norm of Sf ; that is, Sf = sup z∈D (1 − |z| 2 ) 2 |Sf (z)| .…”

Section: Its Most Important Property Is Möbius Invariance: S(t • F )mentioning

confidence: 99%

“…These mappings are obtained by a known process of shearing conformal mappings whose range is convex in one direction. (See [8], Section 3.4 for the shear construction.) Theorem 4.…”

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confidence: 99%

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Schwarzian derivative criteria for valence of analytic and harmonic mappings

Chuaqui¹,

Duren²,

Osgood³

2007

Math. Proc. Camb. Phil. Soc.

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Abstract. For analytic functions in the unit disk, general bounds on the Schwarzian derivative in terms of Nehari functions are shown to imply uniform local univalence and in some cases finite and bounded valence. Similar results are obtained for the Weierstrass-Enneper lifts of planar harmonic mappings to their associated minimal surfaces. Finally certain classes of harmonic mappings are shown to have finite Schwarzian norm. §1. Introduction.The Schwarzian derivative of an analytic locally univalent function f is defined byMore generally, if g is any function analytic and locally univalent on the range of f , thenAs a special case, S(g • T ) = ((Sg) • T )T ′ 2 , since ST = 0 for every Möbius transformation T . For an arbitrary analytic function ψ, the functions f with Schwarzian Sf = 2ψ are those of the form f = w 1 /w 2 , where w 1 and w 2 are linearly independent solutions of the linear differential equation w ′′ + ψw = 0. It follows that Sf = Sg implies f = T • g for some Möbius transformation T . In particular, Möbius transformations are the only functions with Sf = 0. Nehari [14] found that certain estimates on the Schwarzian imply global univalence. Specifically, he showed that if f is analytic and locally univalent in the unit disk D and its Schwarzian satisfies either |Sf (z)| ≤ 2(1 − |z| 2 ) −2 or |Sf (z)| ≤ π 2 /2 for all z ∈ D, then f is univalent in D. Pokornyi [18] then stated, and Nehari [15] proved, that the condition |Sf (z)| ≤ 4(1 − |z| 2 ) −1 also implies univalence. In fact, Nehari [15] unified all three criteria by proving that f is univalent under the The authors are supported by Fondecyt Grant # 1030589.

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“…Further information about harmonic mappings and their relation to minimal surfaces can be found in the book [8].…”

Section: Its Most Important Property Is Möbius Invariance: S(t • F )mentioning

confidence: 99%

Section: Its Most Important Property Is Möbius Invariance: S(t • F )mentioning

confidence: 99%

See 1 more Smart Citation

Schwarzian derivative criteria for valence of analytic and harmonic mappings

Chuaqui¹,

Duren²,

Osgood³

2007

Math. Proc. Camb. Phil. Soc.

Self Cite

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“…For more basic results on harmonic functions one may refer following standard introductory text book by Duren [14], (see also Ahuja [1] and Ponnusamy and Rasila [28,29]). …”

Section: Introductionmentioning

confidence: 99%

Some properties of generalized convolution of harmonic univalent functions

Porwal

Dixit

2013

Demonstratio Mathematica

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Abstract. The purpose of the present paper is to investigate some interesting properties on generalized convolutions of functions for the classes HP * (α), HS(α) and HC(α). Further, an application of the convolution on certain integral operator are mentioned. IntroductionA continuous complex-valued function f = u + iv defined in a simply connected complex domain D is said to be harmonic in D, if both u and v are real harmonic in D. In any simply connected domain we can write f = h+g, where h and g are analytic in D. We call h the analytic part and g the coanalytic part of f. A necessary and sufficient condition for f to be locally univalent and sense-preserving in D is that |h ′ (z)| > |g ′ (z)|, z ∈ D, (see ). For more basic results on harmonic functions one may refer following standard introductory text book by Duren [14], (see also Ahuja [1] and Ponnusamy and Rasila [28,29]Denote by S H the class of functions f = h+g that are harmonic univalent and sense-preserving in the unit disc U = {z : |z| < 1} for which f (0) = f z (0) − 1 = 0. Then, for f = h + g ∈ S H , we may express the analytic functions h and g asA function f (z) of the form (1.1) in S H is said to be harmonic starlike of order α, (0 ≤ α < 1) in U , if and only if (1.2) ∂ ∂θ {arg f (z)} > α, z ∈ U,2000 Mathematics Subject Classification: 30C45, 26D15.

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“…At this point we appeal to a generalized variant of Theorem of Radó-Kneser-Choquet [15], see Theorem 2.11, which ensures that the harmonic extension of f :…”

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confidence: 99%

Mappings of Least Dirichlet Energy and their Hopf Differentials

Iwaniec

Onninen

2013

Arch Rational Mech Anal

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Abstract. The paper is concerned with mappings h : X onto − − → Y between planar domains having least Dirichlet energy. The existence and uniqueness (up to a conformal change of variables in X ) of the energyminimal mappings is established within the class H 2(X, Y) of strong limits of homeomorphisms in the Sobolev space W 1,2 (X, Y) , a result of considerable interest in the mathematical models of Nonlinear Elasticity. The inner variation (of the independent variable in X) leads to the Hopf differential hzhz dz ⊗ dz and its trajectories. For a pair of doubly connected domains, in which X has finite conformal modulus, we establish the following principle:A mapping h ∈ H 2(X, Y) is energy-minimal if and only if its Hopf-differential is analytic in X and real along ∂X. In general, the energy-minimal mappings may not be injective, in which case one observes the occurrence of cracks in X . Nevertheless, cracks are triggered only by the points in ∂Y where Y fails to be convex. The general law of formation of cracks reads as follows:Cracks propagate along vertical trajectories of the Hopf differential from ∂X toward the interior of X where they eventually terminate before making a crosscut.

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Harmonic Mappings in the Plane

Cited by 580 publications

References 107 publications

Schwarzian derivative criteria for valence of analytic and harmonic mappings

Schwarzian derivative criteria for valence of analytic and harmonic mappings

Some properties of generalized convolution of harmonic univalent functions

Mappings of Least Dirichlet Energy and their Hopf Differentials

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