QUASICONFORMAL DEFORMATIONS OF HOLOMORPHIC FUNCTIONS
|
|
- Roland Poole
- 5 years ago
- Views:
Transcription
1 Georgian Mathematical Journal Volume 8 (2), Number 3, QUASICONFORMAL DFORMATIONS OF HOLOMORPHIC FUNCTIONS SAMUL L. KRUSHKAL Dedicated to the memory of N. I. Muskhelishvili on the occasion of his th birthday Abstract. This paper deals with holomorphic functions from Bergman spaces B p in the disk and provides the existence of deformations (variations) which do not increase the norm of functions and preserve some other prescribed properties. Admissible variations are constructed (for even integer p 2) using special quasiconformal maps of the complex plane (in line with a new approach to variational problems for holomorphic functions in Banach spaces). 2 Mathematics Subject Classification: Primary: 3C35, 3C5, 3C62, 32A36. Secondary: 3C75. Key words and phrases: Quasiconformal map, Bergman space, Taylor coefficients, variation, inverse mapping theorem, extremal.. Introduction Quasiconformal maps have now become one of the basic tools in many fields of mathematics and its applications. This paper promotes the development of a new approach to solving variational problems in Banach spaces of holomorphic functions, which recently has been outlined in [5], [6]. The methods adopted in those problems rely mainly on the integral representations of holomorphic functions by means of certain measures, which reduces the problems to the investigation of these measures. This method often encounters great difficulties. Our approach is completely different. It is based on the special kind of quasiconformal deformations of the extended complex plane Ĉ = C { }, which ensure a controlled change of the norm of a given function under variation and also preserve some other prescribed properties of the function. This approach was established in [5], [6] mainly for the Hardy spaces H p in the disk with an even integer p and applied to some well-known conjectures concerning nonvanishing functions. The specific features of these spaces are used there in an essential way. The present paper deals with more general Bergman spaces B p of holomorphic functions. Its purpose is to show the existence of deformations (variations) which do not increase the norm of a function and ISSN X / $8. / c Heldermann Verlag
2 538 SAMUL L. KRUSHKAL preserve some other prescribed properties, for example, vary independent of a suitable number of the Taylor coefficients of this function. Such requirements are, of course, rather rigid. Such problems naturally arise, for example, while studying nonvanishing holomorphic functions. These are closely connected with nonbranched holomorphic coverings. To preserve nonvanishing, we construct quasiconformal deformations of the complex plane which fix the origin. This produces additional rigidity of variations, and we show how it can be resolved in the case of Bergman functions. Recall that the Bergman space B p, < p <, in the unit disk = {z = x + iy C : z < } consists of holomorphic functions f with the finite norm f p = ( f(z) p dxdy ) /p. Let x denote the uclidean norm in R n. 2. Main xistence Theorem The next theorem gives a complete answer to the question on the existence of admissible deformations with controlled distortion of the norm and coefficients of the Bergman functions for even integers p. Theorem. Let m and n be two fixed positive integers (m, n ). For every function f (z) = c + c kz k B 2m H with c and c j ( j < n), k=j which is not a polynomial of degree n n, there exists a number ε > such that for each ε (, ε ) and any points d = (d, d j+,..., d n ) C n j+ and a R satisfying the inequalities d ε, a ε, there is a quasiconformal automorphism h of Ĉ, which is conformal at least in the disk := {w : w < sup f (z) + } and satisfies (a) h() = ; (b) h () = + d ; rom(c) h (k+) () = k! d k, k = j +,..., n; (d) (h F ) (z) 2m 2m F 2m 2m = h (F )f 2m 2m = a, where F (z) = z f (t)dt. The map h can be chosen to have the Beltrami coefficient µ h = w h/ w h with µ h M ε. The quantities ε, M as well as the bound of the remainder in (c) depend only on f, m and n. Proof. We associate with the functions f(z) = c k z k B p the complex Banach space B p of their primitive functions F (z) = z f(t)dt = c k k + zk+, ()
3 letting QUASICONFORMAL DFORMATIONS OF HOLOMORPHIC FUNCTIONS 539 F p := F Bp = f B p. Note that sup F (z) sup f(z), so if f B p H, then F B p H. In the sequel, we use p = 2m. Now define for the annulus = {w : R < w < R + }, with a fixed R sup f (z) +, the integral operators T ρ = ρ(ζ)dξdη ζ w, Πρ = wt = ρ(ζ)dξdη (ζ w) 2, assuming ρ L q (), q 2, and regarding the second integral as a principal Cauchy value. Let us seek the required quasiconformal automorphism h = h µ of Ĉ in the form h(w) = w ρ(ζ)dξdη = w + T ρ(w), (2) ζ w with the Beltrami coefficient µ = µ h supported in, i.e., with µ < κ < and µ(w) = on C\. Substituting (2) into the Beltrami equation w h = µ w h (with w = (/2) ( u i v ), w = (/2) ( u + i v ), w = u + iv), one obtains ρ = µ + µπµ + µπ(µπµ) +. As is well-known, this series is convergent in L q () for some q > 2; thus the distributional derivatives w h = ρ and w h = + Πρ belong to L loc q (C). This implies the estimates ρ Lq( r) M (κ, r, q) µ L (C), Πρ Lq( r) M (κ, r, q) µ L (C); h C(r ) M (κ, r, q) µ in the disks r = {w C : w < r} (r < ) and the smoothness of quasiconformal maps as the functions of parameters: if µ(z; t) is a C -smooth L (C)-function of a real (respectively complex) parameter t, then w h µ(,t) and w h µ(,t) are smoothly R-differentiable (respectively, C-differentiable) L p functions of t; hence, the map t h µ(,t) (z) is C smooth as an element of C( r ). For the proof of these results, which will be exploited below, see, e.g., [8], Ch.2; []; [4], Ch. 2. Setting ν, ϕ = ν(ζ)ϕ(ζ)dξdη, (ν L (), ϕ L ()), one can rewrite (2) for w < R in the form h(w) = w + T µ(w) + ω(w) = w + µ, ϕ k w k + ω(w) (3) k=
4 54 SAMUL L. KRUSHKAL with ϕ k (ζ) = /ζ k+, k =,,... ; ω C(r) M 2 (κ, r) µ 2 (r < ). Comparing (3) for w = F (z) with the above-given conditions (a), (b) and (c), one finds that the following relations are satisfied by the desired Beltrami coefficient µ: µ, ϕ + O( µ 2 ) =, µ, ϕ + O( µ 2 ) = d, µ, ϕ k+ + O( µ 2 ) = d k, k = j,..., n. (4) On the other hand, the condition (d), together with (3), gives the equality (h F ) (z) 2m 2m = (F + T ρ F ) (z) 2m 2m = f (z) + (T µ F ) (z) 2m dxdy + O( µ 2 ) Define = = f (z) (Πµ F ) (z)f (z) 2m dxdy + O( µ 2 ) [ f (z) 2 2 Re[f (z)πµ F (z)f (z)] + Πµ F (z) 2 f (z) 2] m dxdy + O( µ 2 ) = f 2m 2m + 2m [ Re f (z) 2m+ ] µ(ζ)dξdη (ζ F (z)) dxdy + O 2 m ( µ 2 ). φ(ζ) = 2m then the previous equality takes the form f (z) 2m+ dxdy, (5) (ζ F (z)) 2 (h F ) 2m 2m = Re µ, φ + O m ( µ 2 ). (6) The function φ is holomorphic in a domain D Ĉ containing the disk, and φ belongs to the span A 2 of the system {ϕ k } in L 2 (). We also have φ(ζ) in D because for large ζ integral (5) expands to φ(ζ) = b k ζ k with We need a stronger fact. b 2 = 2m c 2m+ >. k=2
5 QUASICONFORMAL DFORMATIONS OF HOLOMORPHIC FUNCTIONS 54 Lemma. Under the assumptions of Theorem, the function φ does not reduce to a linear combination of the fractions ϕ,..., ϕ l, l n, i.e., it cannot be of the form l φ(ζ) = b k ζ k, l n. (7) k= Proof. We modify the arguments exploited in [5]. They rely on the real analyticity of the L p -norm. Assuming the contrary that equality (7) holds, one obtains by (6) that any µ L () with µ ε, ε and such that µ, φ k =, k =,,..., l, satisfies the relation (h F ) 2m 2m = O(ε 2 ). (8) Fix an integer s > n such that c s, and consider in L () the span A l,s of the functions ϕ k, with k l and k > s. Then inf{ ϕ s χ L () : χ A l,s } = d >. By the Hahn Banach theorem, there exists µ L () such that µ, χ =, χ A l,s ; µ, ϕ s =, µ = d. By (2) and (3), the quasiconformal homeomorphism h αµ with α = ε now assumes the form h αµ (w) = w + αw s + O(ε 2 ). (9) After replacing the element ϕ s (w) by ϕ s,β (w) = /(w β) s with small β, we obtain the system ϕ k (w), k =,, 2,..., k s; ϕ s,β (ζ) = /(w β) s, which also constitutes a basis in A 2, and pass to the corresponding map with the same remainder as in (9). Then where h αµ β F (z) := Accordingly, ( h αµ β + [ c s s h αµ β (w) = w + α(w β) s + O(ε 2 ) () C α,β s 2 k z k c k = + α(c ) s + ε = [ c ] s 2 s sαβ(c ) s + z s k zk + ] z s + + O(ε 2 ), α 2 + β 2. ) (z) s 2 F = c k z k + [c s 2 s(s )αβ(c ) s + ]z s 2 () + [c s + sα(c ) s + ]z s + + O(ε 2 ).
6 542 SAMUL L. KRUSHKAL In the disk, let f (z) m = c k,mz k, [(h αµ β F ) (z)] m = Applying Parseval s equality in L 2 (), one obtains (h αµ β c α,β k,m zk. F ) 2m 2m = [(h αµ β F ) ] m 2 2 f m 2 2 ( = rk 2 c α,β k,m 2 c k,m 2) (2) with r 2 k = /(k + ). The right-hand side of (2) is a nonconstant real analytic function of α and β. We have k= c α,β k,m = c k,m + αg k,m (β) = c k,m + O k (α) + O k ( αβ ), where O k (ε)/ε d with the bounds depending only on f. It follows from () that the linear term Cαβ in (2) cannot vanish identically for all small αβ. Therefore (h αµ F ) 2m 2m = O(ε ), O(ε )/ε d, which contradicts (8) for suitable choices of ε = α and β. Denote the scalar product in A 2 by χ, ϕ = χ(z)ϕ(z)dxdy. By Lemma, the power series of φ in the disk R must contain the powers ζ k with k > n. Thus the remainder ψ(ζ) = φ(ζ) (j b k ζ k ) = + b k ζ k, s n +, (3) s does not vanish identically in R. It also satisfies ψ, ϕ k =, k = j +,..., n. Further, for any homeomorphism h = h µ satisfying h() =, with µ L (C) vanishing on C \, we have by (3) the expansion hence, h F (z) = ( + µ, ϕ )F (z) + µ, ϕ k F (z) k + O( µ 2 ); k=2 (h F ) (z) = ( + µ, ϕ )f (z) + k µ, ϕ k F (z) k f (z) k=2 + O( µ 2 ) =: c kz k, µ. k=
7 QUASICONFORMAL DFORMATIONS OF HOLOMORPHIC FUNCTIONS 543 This yields, in particular, that c = c h () and since kf (z) k f (z) = k(c z + one concludes that for j =, and c j j + zj+ + ) k (c + c jz j + ), c j = ( + µ, ϕ )c + O( µ 2 ) c j = ( + µ, ϕ )c j + (j + ) µ, ϕ j+ (c ) j+ + O( µ 2 ) when j >, provided µ is chosen so that µ, ϕ k =, k =,,..., j, and µ, ϕ j+ = µ(ζ)ζ j 2 dξdη. Then c j 2 c j 2 = 2(j + ) Re{d j+ c j( c ) j+ } + O( µ 2 ). Put here µ = tν with µ = and t (t C). Combining with (6) and (7), this results in which shows that c lim j 2 c j 2 t t = 2(j + ) Re{c j( c ) j+ ν, ϕ j+ } = 2(j + ) Re{c j( c ) j+ b j+ ν, ψ =, b j+. (4) We now choose the desired Beltrami coefficient µ of the form with the unknown constants µ = ξ ϕ + ξ ϕ + ξ k ϕ k+ + τ ψ, µ C \ =, (5) k=j ξ, ξ, ξ j,..., ξ n, τ to be determined from equalities (4) and (6). Substituting expression (5) into (4) and taking into account the mutual orthogonality of ϕ k on, one obtains for determining ξ k and τ the nonlinear equations k!d k = ξ k σ 2 k+ + O( µ 2 ), k =,, j +,..., n, (6) with d = and σ k+ = ϕ k+ A 2. A comparison of (5) and (6) gives the equation = Re (h F ) (z) 2m 2m ξ ϕ + ξ ϕ + ξ k ϕ k+ + τ ψ, φ k=j + O( µ 2 ), (7)
8 544 SAMUL L. KRUSHKAL hence the only remaining equation is a relation for Re ξ j, Im ξ j, Re τ, Im τ. Thus we add to (7) three real equations to distinguish a unique solution of the above system. First of all we will seek ξ j satisfying ξ j b j+ σ 2 j+ = ξ b σ 2 ξ b σ 2 k=j+ ξ k b k+ r 2 k+ (8) (which annihilates the main part of the increment of the magnitude n c k 2 after deformation). Then (7) takes the form (h F ) 2m 2m = Re τ ψ, φ + O( µ 2 ), and, letting τ be real, we obtain (h F ) 2m 2m = τκ + O( µ 2 ). (9) Separating the real and imaginary parts in (6), (8) and adding (9), we obtain 2(n j) + 5 real equalities, which define a nonlinear C -smooth (in fact, real analytic) map of the points y = W (x) = W ()x + O( x 2 ) x = (Re ξ, Im ξ, Re ξ, Im ξ, Re ξ j, Im ξ j +,..., Re ξ n, Im ξ n, τ) in a small neighborhood U of the origin in R 2(n j)+5, taking the values y = ( Re d, Im d, Re d, Im d, Re d j+, Im d j+,..., Re d n, Im d n, ) (h F ) 2m 2m also near the origin of R 2(n j)+5. Its linearization y = W ()x defines a linear map R 2(n j)+5 R 2(n j)+5 whose Jacobian differs from the product rr 2 2 n rk+ 2 by a nonzero constant factor. j Therefore, x W ()x is a linear isomorphism of R 2(n j)+5 onto itself, and one can apply to W the inverse mapping theorem. It implies the existence of a Beltrami coefficient µ of the form (5), for which we have the assertions of Theorem. The relations (4) and (8) show a special role played by the first nonzero coefficient c j of f, which cannot be replaced in general by some c k with j < k < n.
9 QUASICONFORMAL DFORMATIONS OF HOLOMORPHIC FUNCTIONS Holomorphic Dependence on Coefficient Parameters d j+,..., d n 3.. As a consequence of the proof of Theorem we obtain Theorem 2. For a fixed small a R, the Beltrami coefficient µ determined by (5) can be chosen to be a complex L -holomorphic function of the parameters d, d j+,..., d n. Proof. This follows from the uniqueness of solutions of the system (6), (8), (9). Let d < ε and a < ε satisfy the assumptions of Theorem and define the desired µ by (5). Fix a such a and a τ = τ > found in the proof of Theorem and solve for this µ = ξ ϕ + ξ ϕ + ξ k ϕ k+ + τ ψ the complex system (6), (8) separately. We already know that it has a unique solution ξ, ξ j+,..., ξ n, and by the inverse mapping theorem these ξ k, hence µ, depend on the given d holomorphically. In particular, the value d j = h (j) ()/j! also moves holomorphically, simultaneously with the independent parameters d j+,..., d n, and one can use by variations the openness of holomorphic maps If we need to construct a quasiconformal homeomorphism h satisfying k=j h (F )f 2m 2m = f 2m 2m, i.e., for a =, then Lemma again allows us to seek a Beltrami coefficient µ of the form (5) with unknown constants ξ, ξ j+,..., ξ n, τ to be determined from equations (4) and µ, φ =. In this case, the existence of a desired µ of such form is again ensured, for small ε, by the inverse mapping theorem, and moreover, this theorem provides also L holomorphy of τ and µ in all parameters d j, d j+,..., d n, which move independently A modification of Theorem. Omitting the assertion (a) of Theorem (i.e., that h() = ), one can drop the requirement for the original function f to be bounded in. Indeed, for f B p with p > 2, the Hölder inequality yields f 2 (p 2)/p f p ; and hence the image domain F (), where f is the primitive function () for f, has a finite area (counting with multiplicity). Therefore this domain must have the exterior points. Now, conjugating the desired quasiconformal homeomorphism h with a linear fractional transformation γ of Ĉ, one reduces the proof to the case, where the primitive function F (z) = z f (t)dt is bounded in the unit disk, and this is,
10 546 SAMUL L. KRUSHKAL in fact, exactly what was used in the proof of Theorem. As a result, the homeomorphism h obtained is conformal in some domain containing F (). 4. Infinitesimal Deformations The following infinitesimal version of Theorem admits a simpler proof. Theorem 3. Given a function f (z) = c kz k B 2m H, distinct from a polynomial of degree n n, then for sufficiently small ε > and ε (, ε ) we have that for every point d = (d 2, d 3,..., d n ) C n and a R so that d ε, a ε, there is a quasiconformal homeomorphism h of Ĉ, which is conformal in the disk D and satisfies: (a) h() =, h () = ; (b) h (k) () = k! d k + O(ε 2 ), k = 2, 3,..., n; (c) h (F )f 2m 2m = a + O(ε 2 ), where again F (z) = z f (t)dt. The Beltrami coefficient of h is estimated similar to Theorem. Proof. By assumption, f (z) contains at least one nonzero term c sz s with s > n. Split the space A 2 into the span and its orthogonal complement k= A l = ϕ, ϕ,..., ϕ l A l = {g A 2 : g, ϕ =, ϕ A l } for < l n and observe that A l contains all convergent series g(ζ) = g k ζ k in the disk R, with g s. Any such g determines the map k=s h ε,s [g](w) := w ε(t ḡ) (s ) (w) = w ε(s )! g(ζ)dξdη (ζ w) s (2) if ε C is chosen close to. Similarly to (3), the restriction of T ḡ onto the disk R = { w < R} assumes the form and thus in (2) T ḡ(w) = g(ζ)dξdη ζ w = ḡ k σ kw 2 k (2) k=s (T ḡ) (s ) (w) = ḡ k σk 2 k(k ) (k s + 2) w k s+. k=s It follows from (2) that (h ε,s [g] F ) (z) 2m 2m = Re ḡ, φ s + O(ε 2 ), ε, (22)
11 with QUASICONFORMAL DFORMATIONS OF HOLOMORPHIC FUNCTIONS 547 φ s (ζ) = 2ms! f (z) 2m+ dxdy. (23) (ζ F (z)) s+ We first verify that as in Lemma, φ s cannot be a linear combination only of ϕ,..., ϕ l for some l n, i.e., the equality φ s (ζ) = l b k, l n, (24) ζk+ cannot take place. Assume that (24) holds, then the relation (22) must reduce to Define (h ε,s [g] (F )f 2m 2m = O(ε 2 ), ε. (25) I(f ; g) = f (z) 2m[ ] (z)dxdy (T ḡ) (s ) F = f (z) 2m+ (T ḡ) (s) F (z)dxdy. Using (2), this integral is evaluated as follows: I(f ; g) = 2 f (re iθ ) 2m[ (T ḡ) (s ) F ] (re iθ )rdrdθ = ( r rdr f (z) m 2 ) m f z z =r ḡ k σk 2 k(k ) (k s + 2)(k s + )F (z) k s f (z)dθ k=s = ( ) m ( rdr c i kz k c kr 2k ) m z =r k= k= z k ( ḡ k σk 2 c ) k s k k (k s + ) k=s k= k zk c kz k dz k= z [ ( = 2ḡ s σss! 2 ) m c k 2 r 2k ] + ḡ k B k (r) rdr. k= k=s+ This shows that by suitable choice of the numbers g s, g s+,..., the corresponding function g A l satisfies I(f ; g) >, and then h ε,s [g] F 2m 2m F 2m 2m = 2 Re { εi(f ; g) } + O(ε) = O(), ε. ε ε
12 548 SAMUL L. KRUSHKAL This shows that equalities (24) and (25) cannot occur for sufficiently small ε and thus the function ψ s (ζ) := φ s (ζ) ϕ k (ζ) = b k ζ k, n n +, k=n does not vanish identically in R. We now construct a linear functional L on the space A 2 whose kernel is the space A n,s, the orthogonal complement of A n,s = ϕ, ϕ,..., ϕ n, ψ s. Fix a complex number d with < d < ε, and first define a linear functional L on A 2 satisfying with given d 2,..., d n and Noting that all ϕ A 2 L (ϕ ) =, L (ϕ k ) = d k, k =, 2,..., n, L (ϕ) =, ϕ A n = ϕ, ϕ 2,..., ϕ n. are of the form ϕ = η k r k ϕ k and by Parseval s equality ϕ 2 A 2 = η k 2, the norm of L can be estimated (using also Schwarz s inequality) by L = sup ϕ A2 = = sup η k 2 = L (ϕ) η k d k /r k k= sup η k 2 = d k 2 /rk 2 ηk 2 < M 2 (n)ε. There exists a function g A 2 representing L so that L (ϕ) = ḡ, ϕ and g 2 = L = O(ε), which defines a polynomial map h : D R C by Let and put h (w) = w g (ζ)dξdη ζ w h F (z) = = w ḡ, ϕ k w k = c k, k zk, d k w k. a = 2 c 2 c k, 2 c Re d + k 2. (26) k=2 k 2 Now define another linear functional L 2 on A 2, setting L 2 (ψ s ) = a a
13 QUASICONFORMAL DFORMATIONS OF HOLOMORPHIC FUNCTIONS 549 and L 2 (ϕ) = for ϕ ψ s and extending by the Hahn Banach theorem from the space ψ s ψ s, where ψ s denotes the span {tψ s : t C} of ψ s onto A 2. Its norm is estimated by L 2 M 3 (/δ)ε, δ = dist(ψ s, ψ s ) >. The corresponding function g 2 A 2 for which L 2 (ϕ) = ḡ 2, ϕ, determines a holomorphic map D R C by (s )! h 2 (w) = g 2 2 = L 2 = O(ε), g 2 (ζ)dξdη (ζ w) s = ḡ 2, ϕ k k(k ) (k s + 2) w k s+. k=s Define finally the map h = h + h 2 : Ĉ Ĉ corresponding to L = L + L 2. This map is holomorphic in the disk R, and it follows from above that h() =, h () = + O(ε), h (k) () = k!d k + O(ε 2 ), k = 2, 3,..., n, with given d k, and (h F ) 2m 2m = a + O(ε 2 ). Restricting h to a smaller disk R = { w R < R}, we get h (w) R M 4 (R )ε, h (j) (w) R M 4 (R )ε, j = 2, 3,..., n. (27) If ε is sufficiently small, h R is univalent in this disk and admits quasiconformal extensions across the circle { w = R } onto the whole complex plane. One can use, for example, its Ahlfors Weill extension [2] with the harmonic Beltrami coefficient µ h (w) = ( R 2 ) 2 S ( w 2 R) 2 2 h, w > R w w 4, where S h = (h /h ) (h /h ) 2 /2 denotes the Schwarzian derivative of h on R. It follows from (27) that µ h = O(ε) as ε. The quasiconformal homeomorphism h constructed above satisfies all the assertions of the theorem except h () =. To establish the latter property, observe that the parameter d has been chosen arbitrarily in the above proof, and take now those values of d for which equality (26) is reduced to a =.
14 55 SAMUL L. KRUSHKAL One obtains a linear equation for the ε-linear term of Re d by which this quantity is determined uniquely. On the other hand, the value of Im d does not affect the linear part of the right-hand side of (26). The corresponding homeomorphism h satisfies h () = + O(ε 2 ), and it remains to rescale h by passing to h(z)/h (). This completes the proof of Theorem xtremal Holomorphic Covering Maps 5.. We present an application of the above theorems to a well-known coefficient problem for holomorphic functions. Let F (z) = C k z k be locally injective in the unit disk. Then its derivative F (z) = f(z) does not vanish in ; thus F is a nonbranched holomorphic covering map F () normalized by F () =. Assume that F B 2m and consider the extremal problem to determine max C n on the closed unit ball B ( B 2m ) = {F (z) = C k z k B 2m : F }. It is equivalent to the Hummel Scheinberg Zalcman problem for nonvanishing Bergman holomorphic functions f in the disk (see [3]). The first coefficient can be estimated in a standard way using Schwarz s lemma for holomorphic maps B ( B 2m ). It gives, together with Parseval s equality applied to f(z) m, that C /. The equality takes place only for the function F (z) = ɛz, ɛ =. However there is no such connection for higher coefficients. Boundedness in the B p -norm yields, by the mean value inequality, the compactness in the topology of locally uniform convergence on, which, in turn, implies the existence of extremal covering maps F maximizing C n on B ( B 2m ). Note that F 2m = since otherwise max C n would increase by passing from F to ( + r)f with appropriate r >. Theorem 4. Any extremal map F maximizing C n is of the form F (z) = Cz + Ckz k, k=n i.e., it satisfies C 2 = = C n = unless F is a polynomial of degree n (with nonvanishing derivative in ). Proof. Let F be different from a polynomial of degree at most n. Then its Taylor series in contains nonzero powers Cs with s > n. We assume the contrary that Cj for some < j < n and reach, by applying Theorem, a contradiction.
15 QUASICONFORMAL DFORMATIONS OF HOLOMORPHIC FUNCTIONS 55 Observe that for any holomorphic function h in a domain D containing all the values F (z) for z, we have h(w) = d k w k in a neighborhood of the origin w =, and ( h F (z) = d Ckz k ) 2 ( + d 2 Ckz k ) n + + dn Ckz k + =: C kz k. Denoting F (z) = f (z) = c kz k with c k = (k + )Ck+, we get where (h F ) (z) = d kckz k + 2d 2 Ckz k kckz k + ( ) n + nd n Ckz k kckz k + = d c k z k + 2d 2 + nd n ( =: c kz k, c k k zk ) n c k k zk c k z k + c k z k + c = d c, c = d c + 2d 2 (c ) 2, c 2 = d c 2 + 3d 2 c c + 3d 3 (c ) 3,.... (28) If h is, in addition, homeomorphic (thus conformal) on F (), then the map h F is a local conformal homeomorphism of the disk. Now, let k = j > and s n be the least indices of nonzero coefficients Ck for < k < n and for k > n, respectively. We apply Theorem starting with j = j and a =, which provides a quasiconformal homeomorphism h of Ĉ satisfying the previous conformality assumptions and obtain from (28) that the coefficient Cn becomes a holomorphic function of the independent parameters d, d j +,..., d n+ defining h, because d j, in itself, is a holomorphic function of these parameters. The openness of holomorphic maps allows us to choose these parameters (at least d and d n ) so that C n = d C n + + d n (C ) n > C n + O(ε) > C n. (29) On the other hand, we have (h F ) 2m = f 2m + O(ε 2 ) = + O(ε 2 ),
16 552 SAMUL L. KRUSHKAL hence the map h F + O(ε 2 ) also belongs to B ( B 2m ). But then (29) violates the extremality of F for C n by a suitable choice of ε >. This proves Cj =. Applying Theorem successively for j = 2, 3,..., n, one obtains the assertion of Theorem 4. It seems likely that similar to what was established in [7] for the Hardy functions f H p (), the extremal map F in B ( B 2m ) cannot reduce to a polynomial of degree n n either The results of this paper extend to arbitrary Hilbert spaces of holomorphic functions in the disk whose norm is real analytic. References. L. V. Ahlfors and L. Bers, Riemann s mapping theorem for variable metrics. Ann. Math. 72(96), L. V. Ahlfors and G. Weill, A uniqueness theorem for Beltrami equations. Proc. Amer. Math. Soc. 3(962), J. A. Hummel, S. Scheinberg, and L. Zalcman, A coefficient problem for bounded nonvanishing functions. J. Anal. Math. 3(977), S. L. Krushkal, Quasiconformal mappings and Riemann surfaces. Wiley, New York, S. L. Krushkal, Quasiconformal maps decreasing the L p norm. Siberian Math. J. 4(2), S. L. Krushkal, Quasiconformal homeomorphisms and Hardy functions. Preprint (2). 7. T. J. Suffridge, xtremal problems for nonvanishing H p functions. Computational Methods and Function Theory, Proceedings, Valparaiso 989 (St. Ruscheweyh,. B. Staff, L.C. Salinas, R. S. Varga, eds.), Lecture Notes in Math. 435, 77 9, Springer, Berlin, I. N. Vekua, Generalized analytic functions. (Russian) Fizmatgiz, Moscow, 959; nglish translation: International Series in Pure and Applied Mathematics 25, Pergamon, New York, 962. (Received 7.5.2) Author s address: Research Institute for Mathematical Sciences Department of Mathematics and Computer Science Bar-Ilan University, 529 Ramat-Gan Israel -mail: krushkal@macs.biu.ac.il
Quasi-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 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 informationMORE NOTES FOR MATH 823, FALL 2007
MORE NOTES FOR MATH 83, FALL 007 Prop 1.1 Prop 1. Lemma 1.3 1. The Siegel upper half space 1.1. The Siegel upper half space and its Bergman kernel. The Siegel upper half space is the domain { U n+1 z C
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 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 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 informationarxiv: v2 [math.ds] 9 Jun 2013
SHAPES OF POLYNOMIAL JULIA SETS KATHRYN A. LINDSEY arxiv:209.043v2 [math.ds] 9 Jun 203 Abstract. Any Jordan curve in the complex plane can be approximated arbitrarily well in the Hausdorff topology by
More informationSZEGÖ ASYMPTOTICS OF EXTREMAL POLYNOMIALS ON THE SEGMENT [ 1, +1]: THE CASE OF A MEASURE WITH FINITE DISCRETE PART
Georgian Mathematical Journal Volume 4 (27), Number 4, 673 68 SZEGÖ ASYMPOICS OF EXREMAL POLYNOMIALS ON HE SEGMEN [, +]: HE CASE OF A MEASURE WIH FINIE DISCREE PAR RABAH KHALDI Abstract. he strong asymptotics
More informationMATH 566 LECTURE NOTES 6: NORMAL FAMILIES AND THE THEOREMS OF PICARD
MATH 566 LECTURE NOTES 6: NORMAL FAMILIES AND THE THEOREMS OF PICARD TSOGTGEREL GANTUMUR 1. Introduction Suppose that we want to solve the equation f(z) = β where f is a nonconstant entire function and
More informationPropagation of Smallness and the Uniqueness of Solutions to Some Elliptic Equations in the Plane
Journal of Mathematical Analysis and Applications 267, 460 470 (2002) doi:10.1006/jmaa.2001.7769, available online at http://www.idealibrary.com on Propagation of Smallness and the Uniqueness of Solutions
More informationTheorem 1. Suppose that is a Fuchsian group of the second kind. Then S( ) n J 6= and J( ) n T 6= : Corollary. When is of the second kind, the Bers con
ON THE BERS CONJECTURE FOR FUCHSIAN GROUPS OF THE SECOND KIND DEDICATED TO PROFESSOR TATSUO FUJI'I'E ON HIS SIXTIETH BIRTHDAY Toshiyuki Sugawa x1. Introduction. Supposebthat D is a simply connected domain
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 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 informationAccumulation constants of iterated function systems with Bloch target domains
Accumulation constants of iterated function systems with Bloch target domains September 29, 2005 1 Introduction Linda Keen and Nikola Lakic 1 Suppose that we are given a random sequence of holomorphic
More informationCONFORMAL MODELS AND FINGERPRINTS OF PSEUDO-LEMNISCATES
CONFORMAL MODELS AND FINGERPRINTS OF PSEUDO-LEMNISCATES TREVOR RICHARDS AND MALIK YOUNSI Abstract. We prove that every meromorphic function on the closure of an analytic Jordan domain which is sufficiently
More informationON INFINITESIMAL STREBEL POINTS
ON ININITSIMAL STRBL POINTS ALASTAIR LTCHR Abstract. In this paper, we prove that if is a Riemann surface of infinite analytic type and [µ] T is any element of Teichmüller space, then there exists µ 1
More informationBloch radius, normal families and quasiregular mappings
Bloch radius, normal families and quasiregular mappings Alexandre Eremenko Abstract Bloch s Theorem is extended to K-quasiregular maps R n S n, where S n is the standard n-dimensional sphere. An example
More informationarxiv:math/ v1 [math.fa] 26 Oct 1993
arxiv:math/9310217v1 [math.fa] 26 Oct 1993 ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES M.I.Ostrovskii Abstract. It is proved that there exist complemented subspaces of countable topological
More informationADJOINT OPERATOR OF BERGMAN PROJECTION AND BESOV SPACE B 1
AJOINT OPERATOR OF BERGMAN PROJECTION AN BESOV SPACE B 1 AVI KALAJ and JORJIJE VUJAINOVIĆ The main result of this paper is related to finding two-sided bounds of norm for the adjoint operator P of the
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 informationTRANSLATION INVARIANCE OF FOCK SPACES
TRANSLATION INVARIANCE OF FOCK SPACES KEHE ZHU ABSTRACT. We show that there is only one Hilbert space of entire functions that is invariant under the action of naturally defined weighted translations.
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 informationBERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS
BERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS SHOO SETO Abstract. These are the notes to an expository talk I plan to give at MGSC on Kähler Geometry aimed for beginning graduate students in hopes to motivate
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 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 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 informationHAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM
Georgian Mathematical Journal Volume 9 (2002), Number 3, 591 600 NONEXPANSIVE MAPPINGS AND ITERATIVE METHODS IN UNIFORMLY CONVEX BANACH SPACES HAIYUN ZHOU, RAVI P. AGARWAL, YEOL JE CHO, AND YONG SOO KIM
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 informationINTEGRATION OPERATORS FROM CAUCHY INTEGRAL TRANSFORMS TO WEIGHTED DIRICHLET SPACES. Ajay K. Sharma and Anshu Sharma (Received 16 April, 2013)
NEW ZEALAN JOURNAL OF MATHEMATICS Volume 44 (204), 93 0 INTEGRATION OPERATORS FROM CAUCHY INTEGRAL TRANSFORMS TO WEIGHTE IRICHLET SPACES Ajay K. Sharma and Anshu Sharma (Received 6 April, 203) Abstract.
More informationCOMPLETELY INVARIANT JULIA SETS OF POLYNOMIAL SEMIGROUPS
Series Logo Volume 00, Number 00, Xxxx 19xx COMPLETELY INVARIANT JULIA SETS OF POLYNOMIAL SEMIGROUPS RICH STANKEWITZ Abstract. Let G be a semigroup of rational functions of degree at least two, under composition
More informationON THE NIELSEN-THURSTON-BERS TYPE OF SOME SELF-MAPS OF RIEMANN SURFACES WITH TWO SPECIFIED POINTS
Imayoshi, Y., Ito, M. and Yamamoto, H. Osaka J. Math. 40 (003), 659 685 ON THE NIELSEN-THURSTON-BERS TYPE OF SOME SELF-MAPS OF RIEMANN SURFACES WITH TWO SPECIFIED POINTS Dedicated to Professor Hiroki Sato
More informationThe Theorem of Gauß-Bonnet in Complex Analysis 1
The Theorem of Gauß-Bonnet in Complex Analysis 1 Otto Forster Abstract. The theorem of Gauß-Bonnet is interpreted within the framework of Complex Analysis of one and several variables. Geodesic triangles
More informationComplex Analysis, Stein and Shakarchi Meromorphic Functions and the Logarithm
Complex Analysis, Stein and Shakarchi Chapter 3 Meromorphic Functions and the Logarithm Yung-Hsiang Huang 217.11.5 Exercises 1. From the identity sin πz = eiπz e iπz 2i, it s easy to show its zeros are
More informationON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 7, July 1996 ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES M. I. OSTROVSKII (Communicated by Dale Alspach) Abstract.
More informationMATH 566 LECTURE NOTES 4: ISOLATED SINGULARITIES AND THE RESIDUE THEOREM
MATH 566 LECTURE NOTES 4: ISOLATED SINGULARITIES AND THE RESIDUE THEOREM TSOGTGEREL GANTUMUR 1. Functions holomorphic on an annulus Let A = D R \D r be an annulus centered at 0 with 0 < r < R
More informationPRODUCTS OF TOEPLITZ OPERATORS ON THE FOCK SPACE
PRODUCTS OF TOEPLITZ OPERATORS ON THE FOCK SPACE HONG RAE CHO, JONG-DO PARK, AND KEHE ZHU ABSTRACT. Let f and g be functions, not identically zero, in the Fock space F 2 α of. We show that the product
More informationBounded and integrable quadratic differentials: hyperbolic and extremal lengths on Riemann surfaces
Geometric Complex Analysis edited by Junjiro Noguchi et al. World Scientific, Singapore, 1995 pp.443 450 Bounded and integrable quadratic differentials: hyperbolic and extremal lengths on Riemann surfaces
More informationThe Dirichlet problem and prime ends
The Dirichlet problem and prime ends arxiv:1503.04306v3 [math.cv] 29 Mar 2015 Denis Kovtonyuk, Igor Petkov and Vladimir Ryazanov 31 марта 2015 г. Аннотация It is developed the theory of the boundary behavior
More informationFinite-dimensional spaces. C n is the space of n-tuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a pre-hilbert space, or a unitary space) if there is a mapping (, )
More information2 Simply connected domains
RESEARCH A note on the Königs domain of compact composition operators on the Bloch space Matthew M Jones Open Access Correspondence: m.m.jones@mdx. ac.uk Department of Mathematics, Middlesex University,
More informationON HARMONIC FUNCTIONS ON SURFACES WITH POSITIVE GAUSS CURVATURE AND THE SCHWARZ LEMMA
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 44, Number 5, 24 ON HARMONIC FUNCTIONS ON SURFACES WITH POSITIVE GAUSS CURVATURE AND THE SCHWARZ LEMMA DAVID KALAJ ABSTRACT. We prove some versions of the Schwarz
More informationAn Introduction to Complex Analysis and Geometry John P. D Angelo, Pure and Applied Undergraduate Texts Volume 12, American Mathematical Society, 2010
An Introduction to Complex Analysis and Geometry John P. D Angelo, Pure and Applied Undergraduate Texts Volume 12, American Mathematical Society, 2010 John P. D Angelo, Univ. of Illinois, Urbana IL 61801.
More informationCOMPACT DIFFERENCE OF WEIGHTED COMPOSITION OPERATORS ON N p -SPACES IN THE BALL
COMPACT DIFFERENCE OF WEIGHTED COMPOSITION OPERATORS ON N p -SPACES IN THE BALL HU BINGYANG and LE HAI KHOI Communicated by Mihai Putinar We obtain necessary and sucient conditions for the compactness
More informationCOMPOSITION SEMIGROUPS ON BMOA AND H AUSTIN ANDERSON, MIRJANA JOVOVIC, AND WAYNE SMITH
COMPOSITION SEMIGROUPS ON BMOA AND H AUSTIN ANDERSON, MIRJANA JOVOVIC, AND WAYNE SMITH Abstract. We study [ϕ t, X], the maximal space of strong continuity for a semigroup of composition operators induced
More informationMATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5
MATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5.. The Arzela-Ascoli Theorem.. The Riemann mapping theorem Let X be a metric space, and let F be a family of continuous complex-valued functions on X. We have
More informationSZEGŐ KERNEL TRANSFORMATION LAW FOR PROPER HOLOMORPHIC MAPPINGS
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 44, Number 3, 2014 SZEGŐ KERNEL TRANSFORMATION LAW FOR PROPER HOLOMORPHIC MAPPINGS MICHAEL BOLT ABSTRACT. Let Ω 1, Ω 2 be smoothly bounded doubly connected
More informationMath Homework 2
Math 73 Homework Due: September 8, 6 Suppose that f is holomorphic in a region Ω, ie an open connected set Prove that in any of the following cases (a) R(f) is constant; (b) I(f) is constant; (c) f is
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 informationON A DISTANCE DEFINED BY THE LENGTH SPECTRUM ON TEICHMÜLLER SPACE
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 28, 2003, 315 326 ON A DISTANCE DEFINED BY THE LENGTH SPECTRUM ON TEICHMÜLLER SPACE Hiroshige Shiga Tokyo Institute of Technology, Department of
More informationMasahiro Yanagishita. Riemann surfaces and Discontinuous groups 2014 February 16th, 2015
Complex analytic structure on the p-integrable Teichmüller space Masahiro Yanagishita Department of Mathematics, Waseda University JSPS Research Fellow Riemann surfaces and Discontinuous groups 2014 February
More informationFundamental Properties of Holomorphic Functions
Complex Analysis Contents Chapter 1. Fundamental Properties of Holomorphic Functions 5 1. Basic definitions 5 2. Integration and Integral formulas 6 3. Some consequences of the integral formulas 8 Chapter
More informationAn introduction to some aspects of functional analysis
An introduction to some aspects of functional analysis Stephen Semmes Rice University Abstract These informal notes deal with some very basic objects in functional analysis, including norms and seminorms
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 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 informationDavid E. Barrett and Jeffrey Diller University of Michigan Indiana University
A NEW CONSTRUCTION OF RIEMANN SURFACES WITH CORONA David E. Barrett and Jeffrey Diller University of Michigan Indiana University 1. Introduction An open Riemann surface X is said to satisfy the corona
More informationRandom Conformal Welding
Random Conformal Welding Antti Kupiainen joint work with K. Astala, P. Jones, E. Saksman Ascona 26.5.2010 Random Planar Curves 2d Statistical Mechanics: phase boundaries Closed curves or curves joining
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 informationl(y j ) = 0 for all y j (1)
Problem 1. The closed linear span of a subset {y j } of a normed vector space is defined as the intersection of all closed subspaces containing all y j and thus the smallest such subspace. 1 Show that
More informationNatural boundary and Zero distribution of random polynomials in smooth domains arxiv: v1 [math.pr] 2 Oct 2017
Natural boundary and Zero distribution of random polynomials in smooth domains arxiv:1710.00937v1 [math.pr] 2 Oct 2017 Igor Pritsker and Koushik Ramachandran Abstract We consider the zero distribution
More informationComposition Operators with Multivalent Symbol
Composition Operators with Multivalent Symbol Rebecca G. Wahl University of South Dakota, Vermillion, South Dakota 57069 March 10, 007 Abstract If ϕ is an analytic map of the unit disk D into itself, the
More informationRings With Topologies Induced by Spaces of Functions
Rings With Topologies Induced by Spaces of Functions Răzvan Gelca April 7, 2006 Abstract: By considering topologies on Noetherian rings that carry the properties of those induced by spaces of functions,
More informationMapping problems and harmonic univalent mappings
Mapping problems and harmonic univalent mappings Antti Rasila Helsinki University of Technology antti.rasila@tkk.fi (Mainly based on P. Duren s book Harmonic mappings in the plane) Helsinki Analysis Seminar,
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 informationX.9 Revisited, Cauchy s Formula for Contours
X.9 Revisited, Cauchy s Formula for Contours Let G C, G open. Let f be holomorphic on G. Let Γ be a simple contour with range(γ) G and int(γ) G. Then, for all z 0 int(γ), f (z 0 ) = 1 f (z) dz 2πi Γ z
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 informationRANDOM HOLOMORPHIC ITERATIONS AND DEGENERATE SUBDOMAINS OF THE UNIT DISK
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 0002-9939(XX)0000-0 RANDOM HOLOMORPHIC ITERATIONS AND DEGENERATE SUBDOMAINS OF THE UNIT DISK LINDA KEEN AND NIKOLA
More informationRigidity of harmonic measure
F U N D A M E N T A MATHEMATICAE 150 (1996) Rigidity of harmonic measure by I. P o p o v i c i and A. V o l b e r g (East Lansing, Mich.) Abstract. Let J be the Julia set of a conformal dynamics f. Provided
More informationON THE NORM OF A COMPOSITION OPERATOR WITH LINEAR FRACTIONAL SYMBOL
ON THE NORM OF A COMPOSITION OPERATOR WITH LINEAR FRACTIONAL SYMBOL CHRISTOPHER HAMMOND Abstract. For any analytic map ϕ : D D, the composition operator C ϕ is bounded on the Hardy space H 2, but there
More informationRadially distributed values and normal families, II
Radially distributed values and normal families, II Walter Bergweiler and Alexandre Eremenko Dedicated to Larry Zalcman Abstract We consider the family of all functions holomorphic in the unit disk for
More informationA Class of Univalent Harmonic Mappings
Mathematica Aeterna, Vol. 6, 016, no. 5, 675-680 A Class of Univalent Harmonic Mappings Jinjing Qiao Department of Mathematics, Hebei University, Baoding, Hebei 07100, People s Republic of China Qiannan
More informationSOLUTION TO RUBEL S QUESTION ABOUT DIFFERENTIALLY ALGEBRAIC DEPENDENCE ON INITIAL VALUES GUY KATRIEL PREPRINT NO /4
SOLUTION TO RUBEL S QUESTION ABOUT DIFFERENTIALLY ALGEBRAIC DEPENDENCE ON INITIAL VALUES GUY KATRIEL PREPRINT NO. 2 2003/4 1 SOLUTION TO RUBEL S QUESTION ABOUT DIFFERENTIALLY ALGEBRAIC DEPENDENCE ON INITIAL
More informationQUASINORMAL FAMILIES AND PERIODIC POINTS
QUASINORMAL FAMILIES AND PERIODIC POINTS WALTER BERGWEILER Dedicated to Larry Zalcman on his 60th Birthday Abstract. Let n 2 be an integer and K > 1. By f n we denote the n-th iterate of a function f.
More informationBLOCH FUNCTIONS ON THE UNIT BALL OF AN INFINITE DIMENSIONAL HILBERT SPACE
BLOCH FUNCTIONS ON THE UNIT BALL OF AN INFINITE DIMENSIONAL HILBERT SPACE OSCAR BLASCO, PABLO GALINDO, AND ALEJANDRO MIRALLES Abstract. The Bloch space has been studied on the open unit disk of C and some
More informationPOWER SERIES AND ANALYTIC CONTINUATION
POWER SERIES AND ANALYTIC CONTINUATION 1. Analytic functions Definition 1.1. A function f : Ω C C is complex-analytic if for each z 0 Ω there exists a power series f z0 (z) := a n (z z 0 ) n which converges
More informationBanach Journal of Mathematical Analysis ISSN: (electronic)
Banach J. Math. Anal. 1 (2007), no. 1, 56 65 Banach Journal of Mathematical Analysis ISSN: 1735-8787 (electronic) http://www.math-analysis.org SOME REMARKS ON STABILITY AND SOLVABILITY OF LINEAR FUNCTIONAL
More informationFree Boundary Minimal Surfaces in the Unit 3-Ball
Free Boundary Minimal Surfaces in the Unit 3-Ball T. Zolotareva (joint work with A. Folha and F. Pacard) CMLS, Ecole polytechnique December 15 2015 Free boundary minimal surfaces in B 3 Denition : minimal
More informationUNIFORM APPROXIMATION BY b HARMONIC FUNCTIONS
UNIFORM APPROXIMATION BY b HARMONIC FUNCTIONS JOHN T. ANDERSON Abstract. The Mergelyan and Ahlfors-Beurling estimates for the Cauchy transform give quantitative information on uniform approximation by
More informationRemark on Hopf Bifurcation Theorem
Remark on Hopf Bifurcation Theorem Krasnosel skii A.M., Rachinskii D.I. Institute for Information Transmission Problems Russian Academy of Sciences 19 Bolshoi Karetny lane, 101447 Moscow, Russia E-mails:
More informationHOLOMORPHIC MAPPINGS INTO SOME DOMAIN IN A COMPLEX NORMED SPACE. Tatsuhiro Honda. 1. Introduction
J. Korean Math. Soc. 41 (2004), No. 1, pp. 145 156 HOLOMORPHIC MAPPINGS INTO SOME DOMAIN IN A COMPLEX NORMED SPACE Tatsuhiro Honda Abstract. Let D 1, D 2 be convex domains in complex normed spaces E 1,
More informationNON-DIVERGENT INFINITELY DISCRETE TEICHMÜLLER MODULAR TRANSFORMATION
NON-DIVERGENT INFINITELY DISCRETE TEICHMÜLLER MODULAR TRANSFORMATION EGE FUJIKAWA AND KATSUHIKO MATSUZAKI Abstract. We consider a classification of Teichmüller modular transformations of an analytically
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 informationTakao Akahori. z i In this paper, if f is a homogeneous polynomial, the correspondence between the Kodaira-Spencer class and C[z 1,...
J. Korean Math. Soc. 40 (2003), No. 4, pp. 667 680 HOMOGENEOUS POLYNOMIAL HYPERSURFACE ISOLATED SINGULARITIES Takao Akahori Abstract. The mirror conjecture means originally the deep relation between complex
More informationTESTING HOLOMORPHY ON CURVES
TESTING HOLOMORPHY ON CURVES BUMA L. FRIDMAN AND DAOWEI MA Abstract. For a domain D C n we construct a continuous foliation of D into one real dimensional curves such that any function f C 1 (D) which
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 informationFORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS
Sairaiji, F. Osaka J. Math. 39 (00), 3 43 FORMAL GROUPS OF CERTAIN Q-CURVES OVER QUADRATIC FIELDS FUMIO SAIRAIJI (Received March 4, 000) 1. Introduction Let be an elliptic curve over Q. We denote by ˆ
More informationInequalities for sums of Green potentials and Blaschke products
Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 Inequalities for sums of reen potentials and Blaschke products Igor. Pritsker Abstract We study inequalities for the ima
More informationPROBLEMS. (b) (Polarization Identity) Show that in any inner product space
1 Professor Carl Cowen Math 54600 Fall 09 PROBLEMS 1. (Geometry in Inner Product Spaces) (a) (Parallelogram Law) Show that in any inner product space x + y 2 + x y 2 = 2( x 2 + y 2 ). (b) (Polarization
More informationHYPERBOLIC DERIVATIVES AND GENERALIZED SCHWARZ-PICK ESTIMATES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 132, Number 11, Pages 339 3318 S 2-9939(4)7479-9 Article electronically published on May 12, 24 HYPERBOLIC DERIVATIVES AND GENERALIZED SCHWARZ-PICK
More informationABELIAN COVERINGS, POINCARÉ EXPONENT OF CONVERGENCE AND HOLOMORPHIC DEFORMATIONS
Annales Academiæ Scientiarum Fennicæ Series A. I. Mathematica Volumen 20, 1995, 81 86 ABELIAN COVERINGS, POINCARÉ EXPONENT OF CONVERGENCE AND HOLOMORPHIC DEFORMATIONS K. Astala and M. Zinsmeister University
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 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 informationDaniel M. Oberlin Department of Mathematics, Florida State University. January 2005
PACKING SPHERES AND FRACTAL STRICHARTZ ESTIMATES IN R d FOR d 3 Daniel M. Oberlin Department of Mathematics, Florida State University January 005 Fix a dimension d and for x R d and r > 0, let Sx, r) stand
More information2. Function spaces and approximation
2.1 2. Function spaces and approximation 2.1. The space of test functions. Notation and prerequisites are collected in Appendix A. Let Ω be an open subset of R n. The space C0 (Ω), consisting of the C
More informationNATIONAL UNIVERSITY OF SINGAPORE. Department of Mathematics. MA4247 Complex Analysis II. Lecture Notes Part IV
NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part IV Chapter 2. Further properties of analytic/holomorphic functions (continued) 2.4. The Open mapping
More informationVOLUME INTEGRAL MEANS OF HOLOMORPHIC FUNCTIONS
VOLUME INTEGRAL MEANS OF HOLOMORPHIC FUNCTIONS JIE XIAO AND KEHE ZHU ABSTRACT. The classical integral means of a holomorphic function f in the unit disk are defined by [ 1/p 1 2π f(re iθ ) dθ] p, r < 1.
More informationProper mappings and CR Geometry
Proper mappings and CR Geometry Partially supported by NSF grant DMS 13-61001 John P. D Angelo University of Illinois at Urbana-Champaign August 5, 2015 1 / 71 Definition of proper map Assume X, Y are
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 informationHilbert spaces. 1. Cauchy-Schwarz-Bunyakowsky inequality
(October 29, 2016) Hilbert spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/fun/notes 2016-17/03 hsp.pdf] Hilbert spaces are
More informationTHE L 2 -HODGE THEORY AND REPRESENTATION ON R n
THE L 2 -HODGE THEORY AND REPRESENTATION ON R n BAISHENG YAN Abstract. We present an elementary L 2 -Hodge theory on whole R n based on the minimization principle of the calculus of variations and some
More informationF (z) =f(z). f(z) = a n (z z 0 ) n. F (z) = a n (z z 0 ) n
6 Chapter 2. CAUCHY S THEOREM AND ITS APPLICATIONS Theorem 5.6 (Schwarz reflection principle) Suppose that f is a holomorphic function in Ω + that extends continuously to I and such that f is real-valued
More information