QUASICONFORMAL DEFORMATIONS OF HOLOMORPHIC FUNCTIONS

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