Mapping problems and harmonic univalent mappings
|
|
- Noah Hampton
- 6 years ago
- Views:
Transcription
1 Mapping problems and harmonic univalent mappings Antti Rasila Helsinki University of Technology (Mainly based on P. Duren s book Harmonic mappings in the plane) Helsinki Analysis Seminar, FILE: plharm3-1.tex, , printed: , Antti Rasila () Mapping problems and harmonic mappings / 36
2 1 Introduction 2 Preliminaries Differential operators / z and / z Canonical representation Jacobian and local univalence Connections to quasiconformal mappings Complex dilatation Second complex dilatation Harmonic univalent mappings 3 Mapping problems Generalizing the Riemann mapping theorem Collapsing Boundary behavior Existence theorems Antti Rasila () Mapping problems and harmonic mappings / 36
3 Introduction A real-valued function u(x, y) is harmonic if it satisfies the Laplace equation u = 2 x 2 u + 2 y 2 u = 0. A complex-valued function f (x + iy) = u(x, y) + iv(x, y) from a domain f : D C is harmonic if the two coordinate functions are harmonic. A complex-valued harmonic function is a harmonic mapping if it is univalent (one-to-one). Harmonic mappings in the plane are univalent complex-valued mappings whose real and imaginary parts are not necessarily conjugate i.e. do not need to satisfy the Cauchy Riemann equations. Antti Rasila () Mapping problems and harmonic mappings / 36
4 Compositions of complex-valued harmonic functions If f : D C, g : f (D) C are harmonic functions, g f is not necessarily harmonic. If f : D C is analytic and g : f (D) C is harmonic, then g f is harmonic. If f : D C is harmonic and g : f (D) C is analytic, then g f is not necessarily harmonic. In fact, even a square or the reciprocal of a harmonic function need not be harmonic. Example: f (x, y) = x 2 y 2 + ix. Then Im(f 2 ) = 16x 0. Inverse of a harmonic mapping need not be harmonic. Antti Rasila () Mapping problems and harmonic mappings / 36
5 Example 1 The simplest example of a harmonic mapping that is not necessarily conformal is an affine mapping f (z) = αz + γ + β z, α β. Every composition of a harmonic mapping with an affine mapping is harmonic, i.e. if f is harmonic, then so is αf + γ + β f. Antti Rasila () Mapping problems and harmonic mappings / 36
6 Example 2 Consider the function f (z) = z z2, which maps the open unit disk D onto the region outside a hypocycloid of three cusps in the circle w = 3/2. To verify its univalence, suppose f (z 1 ) = f (z 2 ) for z 1, z 2 D. Then z 2 1 z 2 2 = 2(z 2 z 1 ) or ( z 1 + z 2 )( z 1 z 2 ) = 2(z 2 z 1 ). By taking absolute values on both sides, we see that this is impossible unless z 1 = z 2 because z 1 z 2 < 2. A similar argument shows that f (z) = z 1 n zn is univalent for each n 2. Antti Rasila () Mapping problems and harmonic mappings / 36
7 Figure 1: Image of the unit disk in the mapping f (z) = z z2. Antti Rasila () Mapping problems and harmonic mappings / 36
8 Figure 2: Image of the unit disk in the mapping f (z) = z z4. Antti Rasila () Mapping problems and harmonic mappings / 36
9 Differential operators / z and / z Define z = 1 ( 2 x i ) and y z = 1 ( 2 x + i ), y where z = x + iy. The equation f / z = 0 is just another way of writing the Cauchy Riemann equations. The Laplacian of f is f = 4 2 f z z. Function f with continuous second partial derivatives is harmonic if and only if f / z is analytic. If f is analytic, then f / z = f (z), i.e. the ordinary derivative. Antti Rasila () Mapping problems and harmonic mappings / 36
10 Properties of / z and / z The operators / z and / z are linear and have the usual properties of differential operators. For example, the product and quotient rules hold: g (fg) = f z z + g f z and ( f ) ( = g f z g z f g ) 1 z g 2, and similarly for / z. The two derivatives are connected by the property ( f ) = f z z. The notation f z = f / z and f z = f / z is often more convenient. Antti Rasila () Mapping problems and harmonic mappings / 36
11 Canonical representation Lemma In a simply connected domain D C a complex valued harmonic function f has the representation f = h + ḡ, where h and g are analytic in D. The representation is unique up to an additive constant. Proof. Since f is harmonic, f z is analytic. Let h = f z, where h is analytic in D. Let g = f h. By definition of h and thus g is analytic in D. g z = f z h z = 0 in D, The uniqueness follows from the fact that a function both analytic and anti-analytic is constant. Antti Rasila () Mapping problems and harmonic mappings / 36
12 Canonical representation (continued) For a harmonic mapping f of the unit disk D, it is convenient to choose the additive constant so that g(0) = 0. The representation f = h + ḡ is then unique and is called the canonical representation of f. Antti Rasila () Mapping problems and harmonic mappings / 36
13 Jacobian and local univalence The Jacobian of a function f = u + iv is J f (z) = u x v x = u xv y u y v x = f z 2 f z 2. If f is analytic, then u y v y J f (z) = (u x ) 2 + (v x ) 2 = f (z) 2. For analytic functions, it is a classical result that J f (z) 0 if and only if f is locally univalent at z. Hans Lewy showed in 1936 that this remains true for harmonic mappings. Consequently f is locally univalent and sense-preserving if f z (z) > f z (z), and sense reversing if f z (z) < f z (z). Antti Rasila () Mapping problems and harmonic mappings / 36
14 Connections to quasiconformal mappings For sense-preserving mappings one sees that ( f z f z ) dz dw ( f z + f z ) dz. These sharp inequalities have the geometric interpretation that f maps an infinitesimal circle onto an infinitesimal ellipse with D f = f z + f z [1, ) f z f z as the ratio of the major and minor axes. The quantity D f (z) is called the dilatation of f at the point z. A sense-preserving homeomorphism f is said to be K-quasiconformal, D f (z) K throughout the given region, where K [1, ) is a constant. Antti Rasila () Mapping problems and harmonic mappings / 36
15 Complex dilatation The ratio µ f = f z /f z is called the complex dilatation of f. If f is sense-preserving, then µ f (z) [0, 1). It may be observed that D f (z) K if and only if µ f (z) K 1 K + 1. It follows that a sense-preserving homeomorphism f is quasiconformal if and only if µ f (z) k < 1. The mapping f is conformal if and only if µ f = 0. Antti Rasila () Mapping problems and harmonic mappings / 36
16 Second complex dilatation In the theory of harmonic mappings, the quantity ν f = f z /f z, known as the second complex dilatation, is often more natural than µ f. Since ν f = µ f, it is clear that f is quasiconformal if and only if ν f (z) k < 1. Antti Rasila () Mapping problems and harmonic mappings / 36
17 Lemma Let f : D C be a function with continuous second order partial derivatives with J f (z) > 0. Then f is harmonic if and only if ω = ν f is analytic. Proof. Let ω = ν f = f z /f z. Then ω(z) < 1 in D. Differentiating the equation f z = ωf z with respect to z, one finds f zz = f z z ω + f z ω z. Now if f is harmonic in D, then f z z = 1 4 f = 0, and hence ω z = 0 in D, so ω is analytic. Conversely, if ω is analytic, then f zz = f z z ω. Since ω(z) < 1, we have f z z = 0, and f is harmonic. Antti Rasila () Mapping problems and harmonic mappings / 36
18 Theorem The only harmonic mappings of C onto C are the affine mappings f (z) = αz + γ + β z, where α, β and γ are complex constants and α β. Proof. Let f = h + ḡ map C harmonically onto C. We may assume that f is sense preserving. Now g (z) < h (z), or g (z)/h (z) < 1 for all z C. By Liouville s theorem g (z)/h (z) b for some complex constant b with b < 1. Integration gives g(z) = bh(z) + c. Thus f has the form f = h + c + bh = F h, where F is an (invertible) affine mapping. It follows that h = F 1 f maps C univalently onto C. But h is analytic, so h(z) = αz + β for some complex constants α, β. Antti Rasila () Mapping problems and harmonic mappings / 36
19 Harmonic univalent mappings Let f = h + ḡ with g(0) = 0, where h(z) = a n z n and g(z) = n=0 b n z n. The class of all sense-preserving harmonic mappings of the unit disk with a 0 = b 0 = 0 and a 1 = 1 is denoted by S H. n=0 S H is a normal family; every sequence of functions in S H has a subsequence that converges locally uniformly in D. S H is not a compact family; it is not preserved under passage to locally uniform limits. The limit function is necessarily harmonic in D, but it need not be univalent. The class of functions f S H with g (0) = 0 will be denoted by S 0 H. Antti Rasila () Mapping problems and harmonic mappings / 36
20 Generalizing the Riemann mapping theorem The classical Riemann mapping theorem states that any simply connected plane domain D, which is not the whole plane, can be conformally mapped onto the unit disk. Next we consider generalizations of this result to the harmonic setting. Obviously, existence of harmonic mappings is not in question as every conformal mapping is harmonic. Rather, the problem is that there are far too many such mappings. Hence, the problem is to develop a meaningful way to classify such mappings, hoping that some additional conditions will uniquely determine the mapping. We will start by taking a look to the theory of quasiconformal mappings in the plane. Antti Rasila () Mapping problems and harmonic mappings / 36
21 The measurable Riemann mapping theorem Theorem (e.g. [Iwaniec-Martin, Thm ]) Suppose that µ: D C is a measurable function with µ < 1. Then there exists a quasiconformal mapping g : D C whose complex dilatation (the Beltrami coefficient) µ g is equal to µ a.e. on D. Moreover, every W 1,2 loc is of the form solution f to the Beltrami equation f z (z) = µ(z)f z (z) f (z) = F (g(z)) where F : g(d) C is an analytic function. Antti Rasila () Mapping problems and harmonic mappings / 36
22 The harmonic case (idea) In the light of this result, it seems plausible to assume that the second complex dilatation ω f = f z /f z would play the similar role in classifying the harmonic mappings f of the unit disk onto a given domain D. Suppose that ω : D D is an analytic function. We try to find a sense-preserving harmonic mapping of D onto a simply connected domain with dilatation ω = f z /f z, unique upto to the normalization f (0) = w 0 D and f z (0) > 0. First we note that if ω f 0, then we have the Riemann mapping theorem. Antti Rasila () Mapping problems and harmonic mappings / 36
23 Constant dilatation Suppose that f is a harmonic function with dilatation ω f, and F = αf + β f, β < α, is an affine transformation of f. Then F has the dilatation F z /F z = αω f + β α + βω f. If ω f c, c < 1, then the choice β = α c will make F analytic. It follows that harmonic mappings with constant dilatation will take the form f = h + c h, where h is a conformal mapping. By taking the inverse image of D in the affine mapping z z + c z, and using the Riemann mapping theorem, we obtain a harmonic mapping with the desired properties. Antti Rasila () Mapping problems and harmonic mappings / 36
24 Collapsing 1/4 However, it turns out that given a simply connected domain D, D C, and an analytic function ω : D D there does not necessarily exist a harmonic mapping of D onto D with dilatation ω. The counterexample is the following. Hengartner and Schober have shown that there is no harmonic mapping of the disk onto itself with ω(z) = z. The idea of the proof is show that if f maps D harmonically into itself, and has dilatation ω = z, then the image must have area less than π. Hence it cannot fill the disk. Suppose that f = h + ḡ is an arbitrary sense-preserving harmonic mapping of D, and write h(z) = a n z n and g(z) = n=0 b n z n. n=1 Antti Rasila () Mapping problems and harmonic mappings / 36
25 Collapsing 2/4 Then the area of the image set f (D) is ( fz A = J f (z) dx dy = 2 f z 2) dx dy because = D D D ( h (z) 2 g (z) 2) dx dy = π n 2 a n 2 z 2(n 1) dx dy = n 2 a n 2 1 n ( a n 2 b n 2), n=1 r=0 2π θ=0 r 2(n 1) r dθ dr 1 = 2πn 2 a n 2 r 2(n 1)+1 dr = 2πn2 a n 2 0 2n = πn a n 2. Antti Rasila () Mapping problems and harmonic mappings / 36
26 Collapsing 3/4 Because f has the dilatation ω f (z) = z, from the differential equation f z = ωf z we have f (z) = zh (z), and hence n=1 (n + 1)b n+1 = na n, for all n = 0, 1, 2,.... (3.1) In particular, b 1 = 0. It follows that ( A = π n an 2 (n + 1) b n+1 2) n = π n + 1 a n 2. (3.2) Next we estimate the integral mean We have M 2 (r, f ) 2 = 1 2π 2π 0 M 2 (r, f ) 2 = 1 2π 2π 0 n=1 f (re iθ ) 2 dθ. ( h(re iθ ) 2 + g(re iθ ) 2) dθ = a n 2 r 2n + b n 2 r 2n. n=0 n=1 Antti Rasila () Mapping problems and harmonic mappings / 36
27 Collapsing 4/4 If f (D) D, then M 2 (r, f ) 1 and thus lim M 2(r, f ) 2 = M 2 (1, f ) 2 = r 1 ( an 2 + b n+1 2) 1. n=0 Because ω f (z) = z, we obtain from (3.1) M 2 (1, f ) 2 = ( 1 + n 2 /(n + 1) 2) a n 2 1. This, together with the estimate 2n/(n + 1) < 1 + n 2 /(n + 1) 2 and (3.2) gives A < π 2 M 2(1, f ) 2 π 2, which is a contradiction because the area of the unit disk is equal to π. Hence, the image set cannot be a the whole disk. Antti Rasila () Mapping problems and harmonic mappings / 36
28 Boundary behavior Hopf s lemma Suppose that D is a Jordan domain with smooth boundary, and u : D R is a non-constant harmonic function with a smooth extension to the boundary. If u has a local minimum at some point w D, then its inner normal derivative is strictly positive at this point: ( u/ n)(w) > 0. The proof of Hopf s lemma is based on Harnack s inequality: R r R + r R + r u(0) u(z) u(0), r = z (3.3) R r for a positive harmonic function u : D(0, R) R +. Antti Rasila () Mapping problems and harmonic mappings / 36
29 Proof We may assume that u(w) = 0 and u(z) > 0 for points in D near w. Then u(z) > 0 for some disk D 0 = D(z 0, r) D whose boundary is tangent to D at w. By taking a similarity transformation if necessary, we may assume that z 0 = 0 and w = r. Then, by Harnack s inequality (3.3), It follows that u(x) u(0) r x r + x, 0 < x < r. u(x) u(r) r x By letting x r, we obtain u(0) r + x, 0 < x < r. u u(0) (r) = u (r) > 0. n x 2r Antti Rasila () Mapping problems and harmonic mappings / 36
30 Boundary dilatation of unit modulus We obtain the following result: Theorem Suppose that f : D D = f (D) is a sense-preserving harmonic mapping with C 1 extension to some open arc I D that maps univalently onto a convex arc γ D. Denote by s(θ) the arc length along γ for e iθ I. If ds/dθ 0 at some point z 0 = e iα I, then the dilatation of f has a continuous extension to a sub-arc of I containing z 0 and ω f (z 0 ) < 1. Antti Rasila () Mapping problems and harmonic mappings / 36
31 Proof 1/2 By taking a Möbius transformation if necessary, we may assume that f is a mapping of the upper half-plane onto D and I R with 0 I. After a similarity transformation, we may assume that D is in the upper half-plane, tangent to the real axis and f (0) = 0 = z 0. Then, f has a smooth homeomorphic boundary extension to I and f (I ) is a convex boundary arc of D. Write w = f (z), where z = x + iy and w = u + iv. It is clear that v x (0) = 0, since v has a local minimum at 0. Because ds/dθ(0) 0, it follows that u x (0) > 0, and the Jacobian of f at the origin is J f (0) = u x (0)v y (0) u y (0)v x (0) = u x (0)v y (0). Antti Rasila () Mapping problems and harmonic mappings / 36
32 Proof 2/2 Because v is a positive harmonic function in the upper half-plane that attains its minimum value 0 at the origin, it follows by Hopf s lemma that v y (0) > 0 as it is the inner normal derivative of v at the origin. It follows that J f (0) > 0, or equivalently, h (0) > g (0). This shows that the dilatation ω f = g /h is continuous up to a sub-arc of I containing the origin, and ω f (0) < 1, proving the claim. Antti Rasila () Mapping problems and harmonic mappings / 36
33 A connection to harmonic quasiconformal mappings In particular, we obtain the following generalization of a result of O. Martio [Martio]. Corollary Suppose that f : D D = f (D) is a sense-preserving harmonic mapping onto a convex domain D with a homeomorphic C 1 extension to the boundary such that ds/dθ > 0 for all e iθ D. Then f is quasiconformal. Proof. The dilatation ω g has a continuous boundary extension satisfying ω f (z) < 1 for all z D. Hence ω(z) k for some constant k < 1, and f is quasiconformal in D. Antti Rasila () Mapping problems and harmonic mappings / 36
34 Existence theorems 1/2 Theorem Suppose that D is a bounded simply connected domain, and D is an analytic Jordan curve, and let w 0 D. Let ω : D C be analytic with ω(z) k for some constant k < 1. Then there exists a harmonic mapping f : D D = f (D) with ω f = ω, f (0) = w 0 and f z (0) > 0. Antti Rasila () Mapping problems and harmonic mappings / 36
35 Existence theorems 2/2 Theorem Suppose that D is a bounded simply connected domain, and D is an analytic Jordan curve Γ, and let w 0 D. Let ω : D C be an analytic function. Then there exists a harmonic mapping f : D D with ω f = ω, f (0) = w 0 and f z (0) > 0 such that radial limits belong to Γ for almost every angle θ. ˆf (e iθ ) = lim r 1 f (re iθ ) Antti Rasila () Mapping problems and harmonic mappings / 36
36 References James Clunie and Terry Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A.I. 9 (1984), Peter Duren, Harmonic mappings in the plane, Cambridge University Press, Tadeusz Iwaniec and Gaven Martin, Geometric Function Theory and Non-linear Analysis, Claredon Press, Oxford, Olli Martio, On harmonic quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser A.I. (1968), Antti Rasila () Mapping problems and harmonic mappings / 36
Harmonic Mappings and Shear Construction. References. Introduction - Definitions
Harmonic Mappings and Shear Construction KAUS 212 Stockholm, Sweden Tri Quach Department of Mathematics and Systems Analysis Aalto University School of Science Joint work with S. Ponnusamy and A. Rasila
More informationThe Inner Mapping Radius of Harmonic Mappings of the Unit Disk 1
The Inner Mapping Radius of Harmonic Mappings of the Unit Disk Michael Dorff and Ted Suffridge Abstract The class S H consists of univalent, harmonic, and sense-preserving functions f in the unit disk,,
More information= 2 x y 2. (1)
COMPLEX ANALYSIS PART 5: HARMONIC FUNCTIONS A Let me start by asking you a question. Suppose that f is an analytic function so that the CR-equation f/ z = 0 is satisfied. Let us write u and v for the real
More informationCurvature Properties of Planar Harmonic Mappings
Computational Methods and Function Theory Volume 4 (2004), No. 1, 127 142 Curvature Properties of Planar Harmonic Mappings Martin Chuaqui, Peter Duren and Brad Osgood Dedicated to the memory of Walter
More informationSolutions to Complex Analysis Prelims Ben Strasser
Solutions to Complex Analysis Prelims Ben Strasser In preparation for the complex analysis prelim, I typed up solutions to some old exams. This document includes complete solutions to both exams in 23,
More informationQuasi-conformal maps and Beltrami equation
Chapter 7 Quasi-conformal maps and Beltrami equation 7. Linear distortion Assume that f(x + iy) =u(x + iy)+iv(x + iy) be a (real) linear map from C C that is orientation preserving. Let z = x + iy and
More informationMA3111S COMPLEX ANALYSIS I
MA3111S COMPLEX ANALYSIS I 1. The Algebra of Complex Numbers A complex number is an expression of the form a + ib, where a and b are real numbers. a is called the real part of a + ib and b the imaginary
More informationAero III/IV Conformal Mapping
Aero III/IV Conformal Mapping View complex function as a mapping Unlike a real function, a complex function w = f(z) cannot be represented by a curve. Instead it is useful to view it as a mapping. Write
More informationQuasiconformal maps and the mapping theorem
4 Quasiconformal maps and the mapping theorem Quasiconformal maps form a branch of complex analysis. I found the subject difficult to learn, mainly because I had a hard time appreciating how smooth the
More informationMath 185 Homework Exercises II
Math 185 Homework Exercises II Instructor: Andrés E. Caicedo Due: July 10, 2002 1. Verify that if f H(Ω) C 2 (Ω) is never zero, then ln f is harmonic in Ω. 2. Let f = u+iv H(Ω) C 2 (Ω). Let p 2 be an integer.
More informationPart IB. Complex Analysis. Year
Part IB Complex Analysis Year 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2018 Paper 1, Section I 2A Complex Analysis or Complex Methods 7 (a) Show that w = log(z) is a conformal
More informationMath 185 Fall 2015, Sample Final Exam Solutions
Math 185 Fall 2015, Sample Final Exam Solutions Nikhil Srivastava December 12, 2015 1. True or false: (a) If f is analytic in the annulus A = {z : 1 < z < 2} then there exist functions g and h such that
More informationRadius of close-to-convexity and fully starlikeness of harmonic mappings
Complex Variables and Elliptic Equations, 014 Vol. 59, No. 4, 539 55, http://dx.doi.org/10.1080/17476933.01.759565 Radius of close-to-convexity and fully starlikeness of harmonic mappings David Kalaj a,
More informationUDC S. Yu. Graf ON THE SCHWARZIAN NORM OF HARMONIC MAPPINGS
0 Probl. Anal. Issues Anal. Vol. 5 3), No., 016, pp. 0 3 DOI: 10.15393/j3.art.016.3511 UDC 517.54 S. Yu. Graf ON THE SCHWARZIAN NORM OF HARMONIC MAPPINGS Abstract. We obtain estimations of the pre-schwarzian
More informationStolz angle limit of a certain class of self-mappings of the unit disk
Available online at www.sciencedirect.com Journal of Approximation Theory 164 (2012) 815 822 www.elsevier.com/locate/jat Full length article Stolz angle limit of a certain class of self-mappings of the
More information(z 0 ) = lim. = lim. = f. Similarly along a vertical line, we fix x = x 0 and vary y. Setting z = x 0 + iy, we get. = lim. = i f
. Holomorphic Harmonic Functions Basic notation. Considering C as R, with coordinates x y, z = x + iy denotes the stard complex coordinate, in the usual way. Definition.1. Let f : U C be a complex valued
More informationComplex Analysis Math 185A, Winter 2010 Final: Solutions
Complex Analysis Math 85A, Winter 200 Final: Solutions. [25 pts] The Jacobian of two real-valued functions u(x, y), v(x, y) of (x, y) is defined by the determinant (u, v) J = (x, y) = u x u y v x v y.
More informationSynopsis of Complex Analysis. Ryan D. Reece
Synopsis of Complex Analysis Ryan D. Reece December 7, 2006 Chapter Complex Numbers. The Parts of a Complex Number A complex number, z, is an ordered pair of real numbers similar to the points in the real
More informationA Note on the Harmonic Quasiconformal Diffeomorphisms of the Unit Disc
Filomat 29:2 (2015), 335 341 DOI 10.2298/FIL1502335K Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat A Note on the Harmonic Quasiconformal
More informationf (n) (z 0 ) Theorem [Morera s Theorem] Suppose f is continuous on a domain U, and satisfies that for any closed curve γ in U, γ
Remarks. 1. So far we have seen that holomorphic is equivalent to analytic. Thus, if f is complex differentiable in an open set, then it is infinitely many times complex differentiable in that set. This
More informationMAT389 Fall 2016, Problem Set 4
MAT389 Fall 2016, Problem Set 4 Harmonic conjugates 4.1 Check that each of the functions u(x, y) below is harmonic at every (x, y) R 2, and find the unique harmonic conjugate, v(x, y), satisfying v(0,
More informationThe result above is known as the Riemann mapping theorem. We will prove it using basic theory of normal families. We start this lecture with that.
Lecture 15 The Riemann mapping theorem Variables MATH-GA 2451.1 Complex The point of this lecture is to prove that the unit disk can be mapped conformally onto any simply connected open set in the plane,
More informationAn Investigation on Minimal Surfaces of Multivalent Harmonic Functions 1
General Mathematics Vol. 19, No. 1 (2011), 99 107 An Investigation on Minimal Surfaces of Multivalent Harmonic Functions 1 Hakan Mete Taştan, Yaşar Polato glu Abstract The projection on the base plane
More informationHarmonic Mappings for which Second Dilatation is Janowski Functions
Mathematica Aeterna, Vol. 3, 2013, no. 8, 617-624 Harmonic Mappings for which Second Dilatation is Janowski Functions Emel Yavuz Duman İstanbul Kültür University Department of Mathematics and Computer
More informationQualifying Exam Complex Analysis (Math 530) January 2019
Qualifying Exam Complex Analysis (Math 53) January 219 1. Let D be a domain. A function f : D C is antiholomorphic if for every z D the limit f(z + h) f(z) lim h h exists. Write f(z) = f(x + iy) = u(x,
More information1. The COMPLEX PLANE AND ELEMENTARY FUNCTIONS: Complex numbers; stereographic projection; simple and multiple connectivity, elementary functions.
Complex Analysis Qualifying Examination 1 The COMPLEX PLANE AND ELEMENTARY FUNCTIONS: Complex numbers; stereographic projection; simple and multiple connectivity, elementary functions 2 ANALYTIC FUNCTIONS:
More informationQuasiconformal and Lipschitz harmonic mappings of the unit disk onto bounded convex domains
Quasiconformal and Lipschitz harmonic mappings of the unit disk onto bounded convex domains Ken-ichi Sakan (Osaka City Univ., Japan) Dariusz Partyka (The John Paul II Catholic University of Lublin, Poland)
More informationMATH 452. SAMPLE 3 SOLUTIONS May 3, (10 pts) Let f(x + iy) = u(x, y) + iv(x, y) be an analytic function. Show that u(x, y) is harmonic.
MATH 45 SAMPLE 3 SOLUTIONS May 3, 06. (0 pts) Let f(x + iy) = u(x, y) + iv(x, y) be an analytic function. Show that u(x, y) is harmonic. Because f is holomorphic, u and v satisfy the Cauchy-Riemann equations:
More informationTheorem [Mean Value Theorem for Harmonic Functions] Let u be harmonic on D(z 0, R). Then for any r (0, R), u(z 0 ) = 1 z z 0 r
2. A harmonic conjugate always exists locally: if u is a harmonic function in an open set U, then for any disk D(z 0, r) U, there is f, which is analytic in D(z 0, r) and satisfies that Re f u. Since such
More informationOn quasiconformality and some properties of harmonic mappings in the unit disk
On quasiconformality and some properties of harmonic mappings in the unit disk Ken-ichi Sakan (Osaka City Univ., Japan) Dariusz Partyka (The John Paul II Catholic University of Lublin, Poland) (The State
More informationLecture 14 Conformal Mapping. 1 Conformality. 1.1 Preservation of angle. 1.2 Length and area. MATH-GA Complex Variables
Lecture 14 Conformal Mapping MATH-GA 2451.001 Complex Variables 1 Conformality 1.1 Preservation of angle The open mapping theorem tells us that an analytic function such that f (z 0 ) 0 maps a small neighborhood
More informationQUASICONFORMAL EXTENSION OF HARMONIC MAPPINGS IN THE PLANE
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 38, 2013, 617 630 QUASICONFORMAL EXTENSION OF HARMONIC MAPPINGS IN THE PLANE Rodrigo Hernández and María J Martín Universidad Adolfo Ibáñez, Facultad
More informationIV. Conformal Maps. 1. Geometric interpretation of differentiability. 2. Automorphisms of the Riemann sphere: Möbius transformations
MTH6111 Complex Analysis 2009-10 Lecture Notes c Shaun Bullett 2009 IV. Conformal Maps 1. Geometric interpretation of differentiability We saw from the definition of complex differentiability that if f
More information2. Complex Analytic Functions
2. Complex Analytic Functions John Douglas Moore July 6, 2011 Recall that if A and B are sets, a function f : A B is a rule which assigns to each element a A a unique element f(a) B. In this course, we
More informationMöbius Transformation
Möbius Transformation 1 1 June 15th, 2010 Mathematics Science Center Tsinghua University Philosophy Rigidity Conformal mappings have rigidity. The diffeomorphism group is of infinite dimension in general.
More informationarxiv: v1 [math.cv] 26 Oct 2009
arxiv:0910.4950v1 [math.cv] 26 Oct 2009 ON BOUNDARY CORRESPONDENCE OF Q.C. HARMONIC MAPPINGS BETWEEN SMOOTH JORDAN DOMAINS DAVID KALAJ Abstract. A quantitative version of an inequality obtained in [8,
More informationNorwegian University of Science and Technology N-7491 Trondheim, Norway
QUASICONFORMAL GEOMETRY AND DYNAMICS BANACH CENTER PUBLICATIONS, VOLUME 48 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1999 WHAT IS A DISK? KARI HAG Norwegian University of Science and
More informationComplex Analysis Problems
Complex Analysis Problems transcribed from the originals by William J. DeMeo October 2, 2008 Contents 99 November 2 2 2 200 November 26 4 3 2006 November 3 6 4 2007 April 6 7 5 2007 November 6 8 99 NOVEMBER
More information(x 1, y 1 ) = (x 2, y 2 ) if and only if x 1 = x 2 and y 1 = y 2.
1. Complex numbers A complex number z is defined as an ordered pair z = (x, y), where x and y are a pair of real numbers. In usual notation, we write z = x + iy, where i is a symbol. The operations of
More informationConvolution Properties of Convex Harmonic Functions
Int. J. Open Problems Complex Analysis, Vol. 4, No. 3, November 01 ISSN 074-87; Copyright c ICSRS Publication, 01 www.i-csrs.org Convolution Properties of Convex Harmonic Functions Raj Kumar, Sushma Gupta
More informationarxiv: v1 [math.cv] 17 Nov 2016
arxiv:1611.05667v1 [math.cv] 17 Nov 2016 CRITERIA FOR BOUNDED VALENCE OF HARMONIC MAPPINGS JUHA-MATTI HUUSKO AND MARÍA J. MARTÍN Abstract. In 1984, Gehring and Pommerenke proved that if the Schwarzian
More informationFunctions of a Complex Variable and Integral Transforms
Functions of a Complex Variable and Integral Transforms Department of Mathematics Zhou Lingjun Textbook Functions of Complex Analysis with Applications to Engineering and Science, 3rd Edition. A. D. Snider
More information11 COMPLEX ANALYSIS IN C. 1.1 Holomorphic Functions
11 COMPLEX ANALYSIS IN C 1.1 Holomorphic Functions A domain Ω in the complex plane C is a connected, open subset of C. Let z o Ω and f a map f : Ω C. We say that f is real differentiable at z o if there
More information7.2 Conformal mappings
7.2 Conformal mappings Let f be an analytic function. At points where f (z) 0 such a map has the remarkable property that it is conformal. This means that angle is preserved (in the sense that any 2 smooth
More informationMAT665:ANALYTIC FUNCTION THEORY
MAT665:ANALYTIC FUNCTION THEORY DR. RITU AGARWAL MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY JAIPUR Contents 1. About 2 2. Complex Numbers 2 3. Fundamental inequalities 2 4. Continuously differentiable functions
More informationCHAPTER 1. Preliminaries
CHAPTER 1 Preliminaries We collect here some definitions and properties of plane quasiconformal mappings. Two basic references for this material are the books by Ahlfors [7] andlehto and Virtanen [117],
More informationON POLYHARMONIC UNIVALENT MAPPINGS
ON POLYHARMONIC UNIVALENT MAPPINGS J. CHEN, A. RASILA and X. WANG In this paper, we introduce a class of complex-valued polyharmonic mappings, denoted by HS pλ, and its subclass HS 0 pλ, where λ [0, ]
More informationHartogs Theorem: separate analyticity implies joint Paul Garrett garrett/
(February 9, 25) Hartogs Theorem: separate analyticity implies joint Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ (The present proof of this old result roughly follows the proof
More informationcarries the circle w 1 onto the circle z R and sends w = 0 to z = a. The function u(s(w)) is harmonic in the unit circle w 1 and we obtain
4. Poisson formula In fact we can write down a formula for the values of u in the interior using only the values on the boundary, in the case when E is a closed disk. First note that (3.5) determines the
More informationu = (u, v) = y The velocity field described by ψ automatically satisfies the incompressibility condition, and it should be noted that
18.354J Nonlinear Dynamics II: Continuum Systems Lecture 1 9 Spring 2015 19 Stream functions and conformal maps There is a useful device for thinking about two dimensional flows, called the stream function
More informationIII.2. Analytic Functions
III.2. Analytic Functions 1 III.2. Analytic Functions Recall. When you hear analytic function, think power series representation! Definition. If G is an open set in C and f : G C, then f is differentiable
More informationondary 31C05 Key words and phrases: Planar harmonic mappings, Quasiconformal mappings, Planar domains
Novi Sad J. Math. Vol. 38, No. 3, 2008, 147-156 QUASICONFORMAL AND HARMONIC MAPPINGS BETWEEN SMOOTH JORDAN DOMAINS David Kalaj 1, Miodrag Mateljević 2 Abstract. We present some recent results on the topic
More informationComplex Variables Notes for Math 703. Updated Fall Anton R. Schep
Complex Variables Notes for Math 703. Updated Fall 20 Anton R. Schep CHAPTER Holomorphic (or Analytic) Functions. Definitions and elementary properties In complex analysis we study functions f : S C,
More informationConformal Mappings. Chapter Schwarz Lemma
Chapter 5 Conformal Mappings In this chapter we study analytic isomorphisms. An analytic isomorphism is also called a conformal map. We say that f is an analytic isomorphism of U with V if f is an analytic
More informationComplex Variables and Elliptic Equations
Estimate of hyperbolically partial derivatives of $\rho$harmonic quasiconformal mappings and its applications Journal: Manuscript ID: Draft Manuscript Type: Research Paper Date Submitted by the Author:
More informationCoefficient Estimates and Bloch s Constant in Some Classes of Harmonic Mappings
Bull. Malays. Math. Sci. Soc. (06) 39:74 750 DOI 0.007/s40840-05-038-9 Coefficient Estimates and Bloch s Constant in Some Classes of Harmonic Mappings S. Kanas D. Klimek-Smȩt Received: 5 September 03 /
More information3. 4. Uniformly normal families and generalisations
Summer School Normal Families in Complex Analysis Julius-Maximilians-Universität Würzburg May 22 29, 2015 3. 4. Uniformly normal families and generalisations Aimo Hinkkanen University of Illinois at Urbana
More informationKey to Complex Analysis Homework 1 Spring 2012 (Thanks to Da Zheng for providing the tex-file)
Key to Complex Analysis Homework 1 Spring 212 (Thanks to Da Zheng for providing the tex-file) February 9, 212 16. Prove: If u is a complex-valued harmonic function, then the real and the imaginary parts
More informationON A SUBCLASS OF HARMONIC UNIVALENT FUNCTIONS DEFINED BY CONVOLUTION AND INTEGRAL CONVOLUTION
International Journal of Pure and pplied Mathematics Volume 69 No. 3 2011, 255-264 ON SUBCLSS OF HRMONIC UNIVLENT FUNCTIONS DEFINED BY CONVOLUTION ND INTEGRL CONVOLUTION K.K. Dixit 1,.L. Patha 2, S. Porwal
More informationChapter 9. Analytic Continuation. 9.1 Analytic Continuation. For every complex problem, there is a solution that is simple, neat, and wrong.
Chapter 9 Analytic Continuation For every complex problem, there is a solution that is simple, neat, and wrong. - H. L. Mencken 9.1 Analytic Continuation Suppose there is a function, f 1 (z) that is analytic
More informationAssignment 2 - Complex Analysis
Assignment 2 - Complex Analysis MATH 440/508 M.P. Lamoureux Sketch of solutions. Γ z dz = Γ (x iy)(dx + idy) = (xdx + ydy) + i Γ Γ ( ydx + xdy) = (/2)(x 2 + y 2 ) endpoints + i [( / y) y ( / x)x]dxdy interiorγ
More informationBOUNDARY CORRESPONDENCE UNDER QUASICONFORMAL HARMONIC DIFFEO- MORPHISMS OF A HALF-PLANE
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 3, 25, 159 165 BOUNDARY CORRESPONDENCE UNDER QUASICONFORMAL HARMONIC DIFFEO- MORPHISMS OF A HALF-PLANE David Kalaj and Miroslav Pavlović Prirodno-matematički
More informationON THE GAUSS MAP OF MINIMAL GRAPHS
ON THE GAUSS MAP OF MINIMAL GRAPHS Daoud Bshouty, Dept. of Mathematics, Technion Inst. of Technology, 3200 Haifa, Israel, and Allen Weitsman, Dept. of Mathematics, Purdue University, W. Lafayette, IN 47907
More informationCHAPTER 2. CONFORMAL MAPPINGS 58
CHAPTER 2. CONFORMAL MAPPINGS 58 We prove that a strong form of converse of the above statement also holds. Please note we could apply the Theorem 1.11.3 to prove the theorem. But we prefer to apply the
More informationHeinz Type Inequalities for Poisson Integrals
Comput. Methods Funct. Theory (14 14:19 36 DOI 1.17/s4315-14-47-1 Heinz Type Inequalities for Poisson Integrals Dariusz Partyka Ken-ichi Sakan Received: 7 September 13 / Revised: 8 October 13 / Accepted:
More informationFollow links Class Use and other Permissions. For more information, send to:
COPYRIGHT NOTICE: Kari Astala, Tadeusz Iwaniec & Gaven Martin: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane is published by Princeton University Press and copyrighted,
More informationINDIAN INSTITUTE OF TECHNOLOGY BOMBAY MA205 Complex Analysis Autumn 2012
INDIAN INSTITUTE OF TECHNOLOGY BOMBAY MA205 Complex Analysis Autumn 2012 September 5, 2012 Mapping Properties Lecture 13 We shall once again return to the study of general behaviour of holomorphic functions
More informationProperties of Analytic Functions
Properties of Analytic Functions Generalizing Results to Analytic Functions In the last few sections, we completely described entire functions through the use of everywhere convergent power series. Our
More informationarxiv: v1 [math.cv] 17 Apr 2008
ON CERTAIN NONLINEAR ELLIPTIC PDE AND QUASICONFOMAL MAPPS BETWEEN EUCLIDEAN SURFACES arxiv:0804.785v [math.cv] 7 Apr 008 DAVID KALAJ AND MIODRAG MATELJEVIĆ Abstract. It is proved that every q.c. C diffeomorphism
More informationAbstract. We derive a sharp bound for the modulus of the Schwarzian derivative of concave univalent functions with opening angle at infinity
A SHARP BOUND FOR THE SCHWARZIAN DERIVATIVE OF CONCAVE FUNCTIONS BAPPADITYA BHOWMIK AND KARL-JOACHIM WIRTHS Abstract. We derive a sharp bound for the modulus of the Schwarzian derivative of concave univalent
More informationComplex Analysis MATH 6300 Fall 2013 Homework 4
Complex Analysis MATH 6300 Fall 2013 Homework 4 Due Wednesday, December 11 at 5 PM Note that to get full credit on any problem in this class, you must solve the problems in an efficient and elegant manner,
More informationMATH5685 Assignment 3
MATH5685 Assignment 3 Due: Wednesday 3 October 1. The open unit disk is denoted D. Q1. Suppose that a n for all n. Show that (1 + a n) converges if and only if a n converges. [Hint: prove that ( N (1 +
More informationINTEGRATION WORKSHOP 2004 COMPLEX ANALYSIS EXERCISES
INTEGRATION WORKSHOP 2004 COMPLEX ANALYSIS EXERCISES PHILIP FOTH 1. Cauchy s Formula and Cauchy s Theorem 1. Suppose that γ is a piecewise smooth positively ( counterclockwise ) oriented simple closed
More informationCOMPLEX ANALYSIS AND RIEMANN SURFACES
COMPLEX ANALYSIS AND RIEMANN SURFACES KEATON QUINN 1 A review of complex analysis Preliminaries The complex numbers C are a 1-dimensional vector space over themselves and so a 2-dimensional vector space
More informationLecture Notes on Minimal Surfaces January 27, 2006 by Michael Dorff
Lecture Notes on Minimal Surfaces January 7, 6 by Michael Dorff 1 Some Background in Differential Geometry Our goal is to develop the mathematics necessary to investigate minimal surfaces in R 3. Every
More informationComplex Analysis Important Concepts
Complex Analysis Important Concepts Travis Askham April 1, 2012 Contents 1 Complex Differentiation 2 1.1 Definition and Characterization.............................. 2 1.2 Examples..........................................
More informationNATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part II
NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part II Chapter 2 Further properties of analytic functions 21 Local/Global behavior of analytic functions;
More informationOn harmonic and QC maps
On harmonic and QC maps Vesna Manojlović University of Belgrade Helsinki Analysis Seminar FILE: vesnahelsinkiharmqc110207.tex 2011-2-6, 20.37 Vesna Manojlović On harmonic and QC maps 1/25 Goal A survey
More informationStream lines, quasilines and holomorphic motions
arxiv:1407.1561v1 [math.cv] 7 Jul 014 Stream lines, quasilines and holomorphic motions Gaven J. Martin Abstract We give a new application of the theory of holomorphic motions to the study the distortion
More informationMATH SPRING UC BERKELEY
MATH 85 - SPRING 205 - UC BERKELEY JASON MURPHY Abstract. These are notes for Math 85 taught in the Spring of 205 at UC Berkeley. c 205 Jason Murphy - All Rights Reserved Contents. Course outline 2 2.
More informationResearch Article Hyperbolically Bi-Lipschitz Continuity for 1/ w 2 -Harmonic Quasiconformal Mappings
International Mathematics and Mathematical Sciences Volume 2012, Article ID 569481, 13 pages doi:10.1155/2012/569481 Research Article Hyperbolically Bi-Lipschitz Continuity for 1/ w 2 -Harmonic Quasiconformal
More informationHARMONIC SHEARS OF ELLIPTIC INTEGRALS
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 5, Number, 5 HARMONIC SHEARS OF ELLIPTIC INTEGRALS MICHAEL DORFF AND J. SZYNAL ABSTRACT. We shear complex elliptic integrals to create univalent harmonic mappings
More information8 8 THE RIEMANN MAPPING THEOREM. 8.1 Simply Connected Surfaces
8 8 THE RIEMANN MAPPING THEOREM 8.1 Simply Connected Surfaces Our aim is to prove the Riemann Mapping Theorem which states that every simply connected Riemann surface R is conformally equivalent to D,
More informationComplex Analysis review notes for weeks 1-6
Complex Analysis review notes for weeks -6 Peter Milley Semester 2, 2007 In what follows, unless stated otherwise a domain is a connected open set. Generally we do not include the boundary of the set,
More informationA TALE OF TWO CONFORMALLY INVARIANT METRICS
A TALE OF TWO CONFORMALLY INVARIANT METRICS H. S. BEAR AND WAYNE SMITH Abstract. The Harnack metric is a conformally invariant metric defined in quite general domains that coincides with the hyperbolic
More informationOld and new order of linear invariant family of harmonic mappings and the bound for Jacobian
doi: 10.478/v1006-011-004-3 A N N A L E S U N I V E R S I T A T I S M A R I A E C U R I E - S K Ł O D O W S K A L U B L I N P O L O N I A VOL. LXV, NO., 011 SECTIO A 191 0 MAGDALENA SOBCZAK-KNEĆ, VIKTOR
More informationResearch Article Coefficient Conditions for Harmonic Close-to-Convex Functions
Abstract and Applied Analysis Volume 212, Article ID 413965, 12 pages doi:1.1155/212/413965 Research Article Coefficient Conditions for Harmonic Close-to-Convex Functions Toshio Hayami Department of Mathematics,
More informationSpring 2010 Exam 2. You may not use your books, notes, or any calculator on this exam.
MTH 282 final Spring 2010 Exam 2 Time Limit: 110 Minutes Name (Print): Instructor: Prof. Houhong Fan This exam contains 6 pages (including this cover page) and 5 problems. Check to see if any pages are
More informationThe purpose of this lecture is to present a few applications of conformal mappings in problems which arise in physics and engineering.
Lecture 16 Applications of Conformal Mapping MATH-GA 451.001 Complex Variables The purpose of this lecture is to present a few applications of conformal mappings in problems which arise in physics and
More informationHomework 27. Homework 28. Homework 29. Homework 30. Prof. Girardi, Math 703, Fall 2012 Homework: Define f : C C and u, v : R 2 R by
Homework 27 Define f : C C and u, v : R 2 R by f(z) := xy where x := Re z, y := Im z u(x, y) = Re f(x + iy) v(x, y) = Im f(x + iy). Show that 1. u and v satisfies the Cauchy Riemann equations at (x, y)
More informationDAVID MAPS AND HAUSDORFF DIMENSION
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 9, 004, 38 DAVID MAPS AND HAUSDORFF DIMENSION Saeed Zakeri Stony Brook University, Institute for Mathematical Sciences Stony Brook, NY 794-365,
More informationINTEGRAL MEANS AND COEFFICIENT ESTIMATES ON PLANAR HARMONIC MAPPINGS
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 37 69 79 INTEGRAL MEANS AND COEFFICIENT ESTIMATES ON PLANAR HARMONIC MAPPINGS Shaolin Chen Saminathan Ponnusamy and Xiantao Wang Hunan Normal University
More informationINDEX. Bolzano-Weierstrass theorem, for sequences, boundary points, bounded functions, 142 bounded sets, 42 43
INDEX Abel s identity, 131 Abel s test, 131 132 Abel s theorem, 463 464 absolute convergence, 113 114 implication of conditional convergence, 114 absolute value, 7 reverse triangle inequality, 9 triangle
More informationConformal maps. Lent 2019 COMPLEX METHODS G. Taylor. A star means optional and not necessarily harder.
Lent 29 COMPLEX METHODS G. Taylor A star means optional and not necessarily harder. Conformal maps. (i) Let f(z) = az + b, with ad bc. Where in C is f conformal? cz + d (ii) Let f(z) = z +. What are the
More informationIntroduction to Complex Analysis by Hilary Priestley Unofficial Solutions Manual
Introduction to Complex Analysis by Hilary Priestley Unofficial Solutions Manual MOHAMMAD EHTISHAM AKHTAR IMPERIAL COLLEGE LONDON http://akhtarmath.wordpress.com Dedicated to my Parents ii Preface This
More informationGeometric Complex Analysis. Davoud Cheraghi Imperial College London
Geometric Complex Analysis Davoud Cheraghi Imperial College London May 9, 2017 Introduction The subject of complex variables appears in many areas of mathematics as it has been truly the ancestor of many
More informationSolutions to practice problems for the final
Solutions to practice problems for the final Holomorphicity, Cauchy-Riemann equations, and Cauchy-Goursat theorem 1. (a) Show that there is a holomorphic function on Ω = {z z > 2} whose derivative is z
More informationMATH 566 LECTURE NOTES 1: HARMONIC FUNCTIONS
MATH 566 LECTURE NOTES 1: HARMONIC FUNCTIONS TSOGTGEREL GANTUMUR 1. The mean value property In this set of notes, we consider real-valued functions on two-dimensional domains, although it is straightforward
More informationComplex Analysis Qual Sheet
Complex Analysis Qual Sheet Robert Won Tricks and traps. traps. Basically all complex analysis qualifying exams are collections of tricks and - Jim Agler Useful facts. e z = 2. sin z = n=0 3. cos z = z
More informationWeek 6 notes. dz = 0, it follows by taking limit that f has no zeros in B δ/2, contradicting the hypothesis that f(z 0 ) = 0.
Week 6 notes. Proof of Riemann Mapping Theorem Remark.. As far as the Riemann-mapping theorem and properties of the map near the boundary, I am following the approach in O. Costin notes, though I have
More information