Approximation Properties of Julia Polynomials - Springer Link

Acta Mathematica Sinica, English Series Jul., 2007, Vol. 23, No. 7, pp. 1303–1310 Published online: Apr. 10, 2006 DOI: 10.1007/s10114-005-0730-2 Http://www.ActaMath.com

Approximation Properties of Julia Polynomials Daniyal M. ISRAFILOV

Burcin OKTAY

Department of Mathematics, Faculty of Art and Sciences, Balikesir University, 10100 Balikesir, Turkey E-mail: [email protected] [email protected] Abstract Let G be a finite simply connected domain in the complex plane C, bounded by a rectifiable Jordan curve L, and let w = ϕ0 (z) be the Riemann conformal mapping of G onto D (0, r0 ) := {w : |w| < r0 }, normalized by the conditions ϕ0 (z0 ) = 0, ϕ0 (z0 ) = 1. In this work, the rate of approximation of ϕ0 by the polynomials, defined with the help of the solutions of some extremal problem, in a closed domain G is studied. This rate depends on the geometric properties of the boundary L. Keywords conformal mapping, extremal polynomials, bounded boundary rotation, Dini-smooth boundary MR(2000) Subject Classification 30E10, 41A10, 30C40

1 Introduction and Results Let G be a finite simply connected domain in the complex plane C, bounded by a rectifiable Jordan curve L, and let z0 ∈ G. By the Riemann mapping theorem, there exists a unique conformal mapping w = ϕ0 (z) of G onto D (0, r0 ) := {w : |w| < r0 }, normalized by the conditions ϕ0 (z0 ) = 0, ϕ0 (z0 ) = 1. The number r0 is called the conformal radius of G with respect to z0 . Let ψ0 (w) be the inverse to ϕ0 (z). Let also G− := extL, D := D (0, 1) , T := ∂D, D− := {w : |w| > 1} , and let ϕ be the conformal mapping of G− onto D− with the normalization ϕ (z) > 0. ϕ (∞) = ∞, lim z→∞ z We denote the inverse mapping of ϕ by ψ. Let Lp (L) and E p (G) be the set of all measurable complex valued functions such that |f |p is Lebesgue integrable with respect to arclength and the Smirnov class of analytic functions in G, respectively. Each function f ∈ E p (G), p ≥ 1 has a nontangential limit almost everywhere (a. e) on L, and if we use the same notation for the nontangential limit, then f ∈ Lp (L) . For p ≥ 1, Lp (L) and E p (G) are the Banach spaces with respect to the norm 1/p p |f (z)| |dz| . f E p (G) = f Lp (L) := L If f has a continuous extension to G, then we also set f G := max |f (z)| , z ∈ G . 1

The following extremal principle was given by Julia in [1]. The function (ϕ0 ) p minimizes the norm f Lp (L) , 1 ≤ p < ∞ in the class of all functions f ∈ E p (G) with the normalization f (z0 ) = 1. On the other hand, if Πn is the class of polynomials pn of degree at most n, satisfying the condition Pn (z0 ) = 1, then the extremal problem inf{Pn Lp (L) : Pn ∈ Πn } is minimized by a unique polynomial Qn,p (p > 1), which is called [2] the nth Julia polynomial for the pair (G, z0 ) (see also [3, 4]). Received January 19, 2005, Accepted May 19, 2005

Israfilov D. M. and Oktay B.

1304 1/p

The function (ϕ0 ) and the polynomials Qn,p play a key role in the investigations of the density problems in the Smirnov spaces E p (G). In particular, the convergence of the sequence ∞ 1/p in E p (G) is equivalent to the density of the algebraic polynomials in {Qn,p (z)}n=0 to (ϕ0 ) this space [4]. In this connection, more detailed and useful information can be found in [5, pp. 438–453]. 1 1 Because of (ϕ0 ) 2 −Pn L2 (L) = Pn 2 − (ϕ0 ) 2 L2 (L) , Pn ∈ Πn , which are proved in [6, L (L)

p. 128], the polynomials Qn,2 may also be determined as the polynomial furnishing a minimum 1 to the integral (ϕ0 ) 2 − Pn L2 (L) , Pn ∈ Πn . Starting from the best approximating property of n,p as the polynomials Qn,2 in L2 (L), Pritsker [7] defined the polynomials Q 1 1 n,p Lp (L) = inf{(ϕ0 ) p − Pn p : Pn ∈ Πn } (1) (ϕ0 ) p − Q L (L) z 1 p p and denoting Jn,p (z) := (Qn,p (t)) dt, z ∈ G estimated the rates (ϕ0 ) − Qn,p p → 0, z0

L (L)

ϕ0 − Jn,p G → 0, n → ∞ in case of a domain with the piecewise analytic boundary L. It is n,2 = Qn,2 . clear that Q Earlier, in the case of p = 2, the estimations in this direction were given by Gaier [6] for some smooth domains, based on the results of Rosenbloom and Warschawski [8] and recently by Pritsker [9] for domains with the piecewise analytic boundaries. In case of p = 2 the polynomials Qn,2 (z) admit [10, pp. 304–317] the representation n−1 j=0 pj (z0 )pj (z) Qn,2 (z) = n−1 n ∈ N, z ∈ G 2 , j=0 |pj (z0 )| with respect to the orthogonal polynomials pj (z) over ∂G, which can effectively be determined by the Gram-Schmidt orthogonalization procedure. The results obtained in this direction are valuable for the approximate construction of the conformal mapping function, because the Riemann conformal mapping theorem states only the existence of the conformal mapping and only for some special domains this mapping has an explicit analytical expression. 1 n,p Lp (L) , ϕ0 − Jn,p → 0, n → ∞ in In this work, we estimate the rates (ϕ0 ) p − Q G terms of the geometric properties of some other domains defined below, where the different smoothness properties and approximations of the conformal mappings are traditionally studied [11, Chapter 3]. Note that another well-known method of approximation for the conformal mappings is the method in which the approximation is conducted by the Bieberbach polynomials produced via the Bergman kernel function. The first result in this direction was proved by Keldysh in [12]. The detailed information about the further results can be found in [13–16], [17, Chapter 3], [18–22] and the references therein. We shall use c, c1 , c2 , . . . to denote constants (in general, different in different relations) depending on only the numbers that are not important for the questions concerned and the notation a bwhich means that a ≤ c1 b and b ≤ c2 a with fixed constants c1 and c2 . of the smooth boundary L and Let ψ0 eit 0 ≤ t ≤ 2π be the conformal parametrization

let β (t) be its tangent direction angle at the point ψ0 eit . Definition 1 (See, [11, p. 63]) The domain G is of bounded boundary rotation if β (t) has 2π n bounded variation, i.e. if 0 |dβ (t)| = suptν ν=1 |β (tν ) − β (tν−1 )| < ∞ for all partitions 0 = t0 < t1 < · · · < tn = 2π. Our main results are presented in the following theorems and proved in Section 3. Theorem 1 If G is a finite smooth domain of bounded boundary rotation, then for every 1 n,p Lp (L) ≤ 1c , n ≥ 1 for ε > 0, there exists a constant c = c (ε) > 0 such that (ϕ0 ) p − Q −ε p > 1 and ϕ0 − Jn,p G ≤ Definition 2

c

1 −ε np

, n ≥ 1, for p = 2, 3, . . . .

np

We say that L ∈ B (α, μ), if ω (β, δ) := sup|h|≤δ β (·) − β (· + h)[0,2π] ≤

Approximation Properties of Julia Polynomials

1305

cδ α lnμ 4δ , δ ∈ (0, π], for some parameters α ∈ (0, 1] and μ ∈ [0, ∞), and for a positive constant c independent of δ. In particular, the class B (α, 0), 0 < α < 1 coincides with the class of Lyapunov curves. On the other hand, it can easily be seen that, c every L ∈ B (α, μ) is a Dini-smooth curve, i.e. ω (β, t) dt < ∞ t 0 for some c > 0. In the case of L ∈ B (α, μ), the approximation properties of the extremal polynomials are given in the following theorem: Theorem 2 If G is a finite domain bounded by a curve L ∈ B (α, μ) with α ∈ (0, 1] and μ ∈ [0, ∞), then ⎧ lnμ n ⎪ ⎪ c , α ∈ (0, 1); ⎨ 1 nα n,p Lp (L) ≤ (ϕ0 ) p − Q n = 2, 3, . . . μ+1 ⎪ ln n ⎪ ⎩ c , α = 1, n for p > 1, and ⎧ μ ⎪ ⎪ c ln n , α ∈ (0, 1); ⎨ nα ϕ0 − Jn,p G ≤ n = 2, 3, . . . ⎪ lnμ+1 n ⎪ ⎩ c , α = 1, n for p = 2, 3, . . . . In particular, if L is a Lyapunov curve, i.e. L ∈ B (α, 0), 0 < α < 1, then we have the following corollary: Corollary 1 If L ∈ B (α, 0), 0 < α < 1, then ϕ0 − Jn,2 G ≤ ncα with a constant c > 0. This estimation improves and generalizes the corresponding Gaier’s [6, p. 131] result c ϕ0 − Jn,2 G ≤ α−1/2 , n given in the case of 1/2 < α < 1. 2 Auxiliary Results Definition 3 For g ∈ Lp = Lp (0, 2π), 1 ≤ p < ∞, the function 1/p 2π |g (x + h) − g (x)|p dx ωp (δ) = ωp (g, δ) := sup |h|≤δ

0

is called the integral modulus of continuity for g ∈ Lp (0, 2π). If ωp (g, δ) = O (δ α ) , 0 < α ≤ 1, we say that g belongs to the class Λpα . Definition 4 The function

1

Let G be a domain with a smooth boundary L and let Φp (w) := (ϕ0 ) p [ψ (w)].

1 ω p ((ϕ0 ) p , δ) := sup Φp weih − Φp (w)Lp (T ) := ωp (Φp , δ) , p ≥ 1 |h|≤δ

1

is called the generalized integral modulus of continuity for (ϕ0 ) p ∈ E p (G) . older’s This definition is correct. Indeed, if p10 + q10 = 1 and |h| ≥ 0, then applying H¨ inequality and later taking into account that ϕ0 , ϕ ∈ Lp (L) for every p ≥ 1 [23], we obtain p Φp Lp (T ) = ϕ0 ϕ L1 (L) ≤ ϕ0 Lp0 (L) ϕ Lq0 (L) < ∞. The following result was proved by Pritsker in [7] (in case of p = 2 see also [8]). Theorem 3 Let G be a Jordan domain with a rectifiable boundary L. If p ∈ N , then 1 n,p Lp (L) , ϕ0 − Jn,p ≤ c (ϕ0 ) p − Q n = 1, 2, . . . G

with a constant c > 0 independent of n.


1306

The following theorem holds: Theorem 4 Let G be a finite smooth domain of bounded boundary rotation and let p > 1. 1

Then (ψ0 ) p eit ∈ Λp1 −ε , for every ε > 0. p

Proof Since L is smooth via ([11, pp. 43–44, Theorem 3.2]), we have π arg ψ0 (eit ) = β (t) − t − 2 for the conformal parametrization and 2π it e +w i π log ψ0 (w) = β (t) − t − dt, w ∈ D. 2π 0 eit − w 2

(2)

(3) 1

Differentiating this equality with respect to w and later multiplying by (ψ0 ) p , we get 1 1 eit i (ψ0 ) p (w) 2π π p −1 ψ0 (w) = (ψ0 ) w ∈ D, 2 β (t) − t − 2 dt, π (eit − w) 0 and also 1 1 1 (ψ0 ) p (w) 2π π p −1 ψ0 (w) = − β (t) − t − dt , w ∈ D. (ψ0 ) π 2 eit − w 0

Since the function β (t) − t − π2 eit1−w is periodic, an integration by parts gives 1 1 (ψ0 ) p (w) 2π d(β (t) − t − π2 ) p −1 , w ∈ D. (ψ0 ) ψ0 (w) = π eit − w 0 Denoting 2π 1/p 1 −1 p1 −1 iθ iθ p (ψ Mp r, (ψ0 ) p ψ0 := re ψ re ) dθ 0 0

(4)

0

from (4) we have p 1 1 2π p1 iθ 2π d(β (t) − t − π/2) p p −1 (ψ ) re ψ0 = p Mp r, (ψ0 ) dθ. π 0 0 eit − reiθ 0 By means older’s inequality, we find of H¨ 1

−1

Mpp r, (ψ0 ) p ψ0 pq0 q1 p1 2π 2π 2π 0 0

iθ p0 d(β (t) − t − π/2) 1 | ψ0 re | dθ dθ , ≤ p it iθ π e − re 0 0 0

where 1/p0 + 1/q0 = 1. Since L is smooth, the first integral is finite and hence pq0 1/q0 2π 2π 1 d(β (t) − t − π/2) p p −1 ψ0 ≤ c1 , Mp r, (ψ0 ) dθ eit − reiθ 0 0 or pq0 1/(pq0 ) 2π 2π 1 d(β (t) − t − π/2) p −1 Mp r, (ψ0 ) ψ0 ≤ c2 . dθ eit − reiθ 0 0 Applying the integral variant of Minkowski’s inequality to the right side, we obtain that 1/(pq0 ) 2π 2π 1 dθ p −1 Mp r, (ψ0 ) ψ0 ≤ c2 | d(β (t) − t − π/2) | . (5) | eit − reiθ |pq0 0 0 Taking into account the inequality 2π dθ c3 ≤ pq0 −1 , it − reiθ |pq0 | e (1 − r) 0 which is verified easily, from (5) we get 2π 1 c4 p −1 ψ0 ≤ | d(β (t) − t − π/2) | . Mp r, (ψ0 ) pq −1 0 (1 − r) pq0 0


1307

Since G is a domain of bounded boundary rotation, the function β (t) − t − π/2 also has bounded variation. This property implies that 1 c5 −1 Mp r, (ψ0 ) p ψ0 ≤ . 1− 1 (1 − r) pq0 Choosing the number q0 > 1sufficiently close to 1, wec have 1 5 p −1 Mp r, (ψ0 ) , ψ0 ≤ 1 1−( p −ε) (1 − r) for every ε > 0. Now applying the well known Hardy–Littlewood theorem (see for example: [24, p. 78]) 1 from the last inequality, we deduce that (ψ0 ) p (eit ) ∈ Λp1 −ε . p

Lemma 1 If p > 1 and G is a smooth domain of bounded boundary rotation, then ωp (Φp , 1/n) ≤ 1c−ε for every ε > 0. np

Proof In fact, with the help of H¨ older’s inequality 1/p 1 1

ih ih p p p Φp (we ) − Φp (w) Lp (T ) = − (ϕ0 ) [ψ (w)] | | dw | | (ϕ0 ) ψ we T p 1/p 1 1 − = | dw | 1 1 ) p [ϕ (ψ (weih ))] T (ψ0 (ψ0 ) p [ϕ0 (ψ (w))] 0 1/p (ψ ) p1 [ϕ (ψ (w))] − (ψ ) p1 ϕ ψ weih p 0 0 0 0 = | dw | 1 1 ) p [ϕ (ψ (weih ))] · (ψ ) p [ϕ (ψ (w))] T (ψ0 0 0 0 1/(pp0 ) 1 1

ih pp0 p p | | dw | ≤ | (ψ0 ) [ϕ0 (ψ (w))] − (ψ0 ) ϕ0 ψ we T

1/(pq0 ) | dw | , ih q0 T | (ψ0 ) [ϕ0 (ψ (we ))] · (ψ0 ) [ϕ0 (ψ (w))] | older where 1/p0 + 1/q0 = 1. The finiteness of the last factor is obtained by applying the H¨ inequality several times and taking into account that ϕ , ϕ1 ∈ Lp (L) and ψ0 , ψ1 ∈ Lp (T ) for 0 every p > 1 [23]. Hence we have ωp (Φp , 1/n) = sup Φp (weih ) − Φp (w) Lp (T ) ×

|h|≤1/n

≤ c6 sup

|h|≤1/n

T

1/(pp0 ) 1 1

pp0 | | dw | | (ψ0 ) p [ϕ0 (ψ (w))] − (ψ0 ) p ϕ0 ψ weih ,

1 and by virtue of Theorem 4, ωp (Φp , 1/n) ≤ c7 sup|h|≤1/n | ϕ0 (ψ (w)) − ϕ0 ψ weih | pp0 −ε . Since for a smooth boundary L, the mapping functions ϕ0 and ψ belong to the H¨ older class with exponent 1 − ε for every ε > 0 on L and on T , respectively, from the last inequality we derive that: ωp (Φp , 1/n) ≤ 1c8 −ε . n pp0

Choosing the number p0 > 1 sufficiently close to 1, we get ωp (Φp , 1/n) ≤ ε > 0. Lemma 2

If L ∈ B(α, μ) with α ∈ (0, 1] and μ ∈ [0, ∞), then ⎧ 4 ⎪ ⎨ cδ α lnμ , 1

1 δ ω((ψ0 )1/p , δ): = sup (ψ0 ) p weih − (ψ0 ) p (w) T ≤ ⎪ |h|≤δ ⎩ cδ lnμ+1 4 , δ

Proof If g (t) := arg ψ0 eit , then by (2)

c

1 −ε

np

, for every

α ∈ (0, 1); α = 1.

4 ω (g, δ) := sup g (t + h) − g (t) [0,2π] ≤ ω (β, δ) + ω (t, δ) ≤ c9 δ α lnμ . δ |h|≤δ

(6)


1308

δ ω(g,t)

π ω(g,t)

On the other hand, for ω ∗ (g, δ) := 0 t dt + δ δ t2 dt we have ⎧ 4 ⎪ ⎨ cδ α lnμ , α ∈ (0, 1); δ ω ∗ (g, δ) ≤ (7) ⎪ ⎩ cδ lnμ+1 4 , α = 1. δ Indeed, if α ∈ (0, 1), then for a sufficiently small ε > 0 and a constant c10 independent of δ, we get tα−ε lnμ 4t ≤ c10 δ α−ε lnμ 4δ , t ∈ (0, δ]. In addition, lnμ 4t ≤ lnμ 4δ , t ∈ [δ, π) for every μ ∈ [0, ∞). Hence by the relation (6), we have π α μ4 δ α−ε μ 4 t ln t t ln t dt + c δ dt ω ∗ (g, δ) ≤ c9 9 1−ε t t2 0 δ δ π 4 μ 4 α−ε μ 4 ε−1 ln t dt + c9 δ ln tα−2 dt ≤ cδ α lnμ . ≤ c11 δ δ 0 δ δ δ If α = 1, then by similar arguments we get π μ4 π δ ε μ4 δ t ln t ln t 4 4 ε μ 4 −ε ω ∗ (g, δ) ≤ c12 dt ≤ c dt + c δ δ ln t dt − c δ lnμ d ln 13 14 13 ε t t δ 0 t t 0 δ δ μ 4 μ+1 4 μ+1 4 + c16 δ ln ≤ cδ ln . ≤ c15 δ ln δ δ δ

Since L is Dini-smooth, the 2π periodic function g (t) = arg ψ0 eit = β (t) − t − π2 is Dinicontinuous on the real line. Therefore, applying Proposition 3.4 in [11] to the relation (3), we conclude that the function log ψ0 (w) has a continuous extension to D and furthermore |log ψ0 (w1 ) − log ψ0 (w2 )| ≤ cω ∗ (g, δ) (8) for w1 , w2 ∈ D with |w1 − w2 | ≤ δ < 1. On the other hand, for |ζ1 | ≤ M , |ζ2 | ≤ M , we have ∞ ζ nM n−1 e 1 − eζ2 ≤ |ζ2 − ζ1 | ≤ c (M ) |ζ2 − ζ1 | . n! n=1 In this inequality, setting ζk = p1 log ψ0 (wk ), k = 1, 2, by (8) we obtain 1 p1 |w1 − w2 | ≤ δ < 1. (ψ0 ) (w1 ) − (ψ0 ) p (w2 ) ≤ c17 ω ∗ (g, δ) ,

(9)

Now combining the relations (7) and (9), we obtain ⎧ 4 ⎪ ⎨ cδ α lnμ , α ∈ (0, 1); 1 δ ω (ψ0 ) p , δ ≤ ⎪ ⎩ cδ lnμ+1 4 , α = 1, δ which completes the proof. Lemma 3 If L ∈ B(α, μ) with α ∈ (0, 1] and μ ∈ [0, ∞), then ⎧ 4 ⎪ ⎨ cδ α lnμ , α ∈ (0, 1); δ ω (Φp , δ) ≤ ⎪ ⎩ cδ lnμ+1 4 , α = 1. δ Proof Since L is Dini-smooth, the functions ψ0 and ϕ0 are continuous on D and on G, respectively. The same properties are also valid for the functions ψ and ϕ , respectively on D− and on G− . Moreover, the relations | ψ0 || ψ | 1 on | w |= 1, and | ϕ0 || ϕ | 1 on L hold [25]. Hence

ϕ0 ψ weih − ϕ0 [ψ (w)] ψ weih − ψ (w) weih − w = eih − 1 |h| T T T and by Lemma 2 we get

1 1

ω (Φp , δ) = sup Φp weih − Φp (w) = sup (ϕ0 ) p ψ weih − (ϕ0 ) p [ψ (w)] T

|h|≤δ

= sup |h|≤δ

1 1

(ψ0 ) p

[ϕ0

[ψ (weih )]]

−

|h|≤δ

1 1

(ψ0 ) p

[ϕ0 [ψ (w)]]

T

T


1309

1 1

≤ c18 sup (ψ0 ) p [ϕ0 [ψ (w)]] − (ψ0 ) p ϕ0 ψ weih

T

|h|≤δ

⎧ 4 ⎪ ⎨ cδ α lnμ , δ ≤ ⎪ ⎩ cδ lnμ+1 4 , δ

α ∈ (0, 1); α = 1.

Slightly modifying the proof of Theorem 3 in [21] we obtain the following approximation 1 theorem for (ϕ0 ) p : Theorem 5

Let G be a domain bounded by a smooth Jordan curve L and let p > 1. Then 1

1

(ϕ0 ) p − Sn ((ϕ0 ) p , ·)Lp (L) ≤ c ωp+ε (Φp , 1/n), 1 1 n p for every ε > 0, where Sn ((ϕ0 ) p , z) := k=0 ak ((ϕ0 ) )Fk (z) , n = 0, 1, 2, . . . are the n-th 1 partial sums of the Faber series of (ϕ0 ) p . In the proof of Theorem 2, we shall use the following approximation theorem in E p (G) shown in [26]: Theorem 6 Let f∈ E p (G), 1 < p < ∞. If L is Dini-smooth, then for every natural number

n, ωp f, n1 there is a polynomial Pn (z, f ) of degree at most n such that, f − Pn (z, f )Lp (L) ≤ c

with a constant c > 0 independent of n, where ω p f, n1 := f ◦ ψ weih − f ◦ ψ (w)Lp (T ) . 3

Proof of Main Results

Proof of Theorem 1 Let Qn be a polynomial of degree at most n. 1 1/p is analytic, the function Since the function [(ϕ0 ) p (ψ0 (w)) − Qn (ψ0 (w))]. [ψ0 (w)] 1 p p | (ϕ0 ) (ψ0 (w)) − Qn (ψ0 (w)) | . |ψ0 (w)| is subharmonic in D (0, r0 ). Hence by ([24, p. 7, Theorem 1.4]) p p1 (ϕ0 ) (ψ0 (0)) − Qn (ψ0 (0)) . |ψ0 (0)|

p

1 2π p1 iθ − Qn ψ0 r0 eiθ . ψ0 r0 eiθ dθ ≤ (ϕ0 ) ψ0 r0 e 2π 0 p 1 p1 = (ϕ0 ) (ψ0 (w)) − Qn (ψ0 (w)) . |ψ0 (w)| |dw| 2πr0 ∂D(0, r0 ) p p 1 1 p1 p1 = (ϕ0 ) (z) − Qn (z) |dz| = (ϕ0 ) − Qn p . 2πr0 Γ 2πr0 L (L) 1

On the other hand, since ϕ0 (z0 ) = 1 and z0 = ψ0 (0), choosing the branch of (ϕ0 ) p which takes the value 1 at the point z0 1we get p p 1 p |1 − Qn (z0 )| = (ϕ0 ) p (z0 ) − Qn (z0 ) = (ϕ0 ) p (ψ0 (0)) − Qn (ψ0 (0)) and hence |1 − Qn (z0 )|p ≤

1 1 (ϕ0 ) p − Qn pLp (L) . 2πr0

(10)

n,p . Now we shall use the extremal property of Q Since [Q 1] is equal to1 1 at the point z0 , we obtain n − 1Qn (z0 ) + p ≤ (ϕ0 ) p − [Qn − Qn (z0 ) + 1] (ϕ0 ) − Qn,p Lp (L)

1 ≤ (ϕ0 ) p − Qn 1 ≤ (ϕ0 ) p − Qn

Lp (L)

Lp (L)

Lp (L)

+ 1 − Qn (z0 )

Lp (L)

+ Γ

|1 − Qn (z0 )|p |dz|

1/p ,


1310

and hence by (10)

p1 n,p (ϕ0 ) − Q

Lp (L)

1 ≤ c (ϕ0 ) p − Qn

Lp (L)

.

(11)

According to Theorem 5 and Lemma 1, respectively, we finally get 1 1 1 1 p1 n,p ≤ c ( ϕ0 ) p − Sn (ϕ0 ) p , · p ≤ cωp+ε Φp , ≤ c 1 −ε . (ϕ0 ) − Q p n L (L) L (L) p n Proof of Theorem 2 The proof is similar to the proof of Theorem 1. After the relation (11), the proof is completed using Theorem 6 and Lemma 3 instead of Theorem 5 and Lemma 1, respectively. References [1] Julia, G.: Lecons sur la représentation conforme des aires simplement connexes, Paris, 1931 [2] Keldsyh, M. V., Lavrentiev, M. A.: On the theory of conformal mappings. Dokl. Akad. Nauk SSSR, 1, 85–87 (1935)(Russian) [3] Keldysh, M. V.: On a class of extremal polynomials. Dokl. Akad. Nauk SSSR, 4, 163–166 (1936) (Russian) [4] Keldsyh, M. V., Lavrentiev, M. A.: Sur la representation conforme des domaines limités pur des courbes rectifiables. Ann. Sci. Ecole Norm. Sup., 54, 1–38 (1937) [5] Goluzin, G. M.: Geometric theory of functions of a complex variable, In: Translation of Mathematical Monographs, Vol 26, American Mathematical Society, Providence, RI, 1969 [6] Gaier, D.: Konstructive Methoden der konformen Abbildung, Springer-Verlag, Berlin, 1964 [7] Pritsker, I. E.: Convergence of Julia polynomials. J. d’Analyse Mat., 94, 343–361 (2004) [8] Rosenbloom, P. C., Warschawski, S. E.: Approximation by polynomials, in: “Lectures on functions of a complex variable”, Ann. Arbor, University of Michigan Press, 287–302, 1955 [9] Pritsker, I. E.: Approximation of Conformal Mapping via the Szeg¨ o Kernel Method. Computational Methods and Function Theory, 3(1), 79–94 (2003) [10] Smirnov, V. I., Lebedev, N. A.: Functions of Complex Variable, Constructive Theory, MIT Press, Cambridge, 1968 [11] Pommerenke, Ch.: Boundary behaviour of conformal maps, Springer-Verlag, Berlin, 1992 [12] Keldysh, M. V.: Sur l’approximation en moyenne quadratique des fonctions analytiques. Math. Sb., 5(47), 391–401 (1939) [13] Mergelyan, S. N.: Certain questions of the constructive theory of functions. Trudy Math. Inst. Steklov, 37, 1–91 (1939) (Russian) [14] Wu, X. M.: On Bieberbach polynomials. Acta Math. Sinica, Chinese Series, 13, 145–151 (1963) [15] Suetin, P. K.: Polynomials ortogonal over a region and Bieberbach polynomials, Proc. Steklov Inst. Math., Vol. 100. Providence, RI: American Mathematical Society [16] Andrievskii, V. V.: Convergence of Bieberbach polynomials in domains with piecewise-quasiconformal boundary (Russian), In: Theory of Mappings and Approximation of Functions, Kiev, Naukova Dumka, 3–18, 1983 [17] Andrievskii, V. V.: Pritsker, I. E.: Convergence of Bieberbach polynomials in domains with interior cusps. J. d Analyse Math., 82, 315–332 (2000) [18] Gaier, D.: On the convergence of the Bieberbach polynomials in region with corners. Constr. Approx., 4, 289–305 (1988) [19] Israfilov, D. M.: Approximation by p-Faber polynomials in the weighted smirnov class E p (G, ω) and the Bieberbach polynomials. Constr. Approx., 17, 335–351 (2001) [20] Abdullayev, F. G.: Uniform convergence of the Bieberbach polynomials inside and on the closure of domains in the complex plane. East Journal of Approx., 7(1), 77–101 (2001) [21] Israfilov, D. M.: Uniform convergence of the Bieberbach polynomials in closed smooth domains of bounded boundary rotation. Journal of Approximation Theory, 125, 116–130 (2003) [22] Israfilov, D. M.: Uniform convergence of the Bieberbach polynomials in closed Radon domains. Analysis, 23, 51–64 (2003) [23] Warschawski, S. E.: Recent results in numerical methods of conformal mapping, in: “Procedings of Symposia in Applied Mathematics”. Vol. VI. Numerical Analysis, McGraw-Hill Book Company, Inc., New York, 219– 250, 1956 [24] Duren, P. L.: Theory of H p Spaces, Academic Press, New York, London, 1970. ¨ [25] Warschawski, S. E.: Uber das Randverhalten der Ableitung der Abbildungsfunktion bei konformer Abbildung. Math. Zeitschrift, 35, 321–456 (1932) [26] Alper, S. Ya.: Approximation in the mean of analytic functions of class Ep (Russian), In: Investigations on the modern problems of the function theory of a complex variable, Moscow: Gos. Izdat. Fiz. Mat. Lit., 273–286, 1960