Convergence of the p-Bieberbach polynomials in regions with zero angles

Uniform convergence of the p-Bieberbach polynomials is proved in the case of a simply connected region bounded by a piecewise quasiconformal curve with certain interior zero angles on the corner where two arcs meet.


INTRODUCTION
Finding the Riemann mapping function for a given region is a very famous and important problem for researchers.The reason is that this function has many applications in some branches of mathematics.There are some construction methods of this function for simple regions differing from those for general regions.So, the best way is to approximate this function by using some extremal polynomials.
Let G be a finite region with 0 ∈ G bounded by Jordan curve L := ∂ G and let w = ϕ(z) be a conformal mapping of G onto the disk {w : |w| < r 0 } with ϕ(0) = 0, ϕ (0) = 1, where r 0 is called the conformal radius of G with respect to 0. Denote by A 1 p (G) , p > 0 the set of functions f (z) analytic in G satisfying f (0) = 0, f (0) = 1 such that where dσ z is a two-dimensional Lebesque measure on G, and denote by ℘ n the set of all algebraic polynomials P n (z) of degree at most n, satisfying P n (0) = 0, P n (0) = 1.Consider the following extremal problem: Using a method similar to the one given in [10, p. 137], it is seen that there exists an extremal polynomial P * n (z) furnishing to the problem (1.1).These polynomials P * n (z) are determined uniquely in case p > 1 [10, p. 142].In [14] the solution of (1.1) was called p-Bieberbach polynomials and denoted by B n,p (z), and its approximation properties in uniform norm were investigated, i.e.
where γ depends on the geometric properties of Γ and c is a constant independent from n.
In case p = 2 the p-Bieberbach polynomials coincide with the well-known n-th Bieberbach polynomial B n (z) ≡ B n,2 (z) (see, for example, [13,20]).Approximation properties in the uniform norm of B n (z) on G were first observed by Keldysh in [13] for the domains with sufficiently smooth boundaries.A considerable progress in this area was achieved in [2,4,6,7,11,12,16,18,20] and others.
In [14], approximation properties of p-Bieberbach polynomials were investigated in case the region was bounded by a quasiconformal curve.It is well known that these curves do not allow interior zero angles.The effect of zero angles for these extremal polynomials has not yet been studied but our results show this effect.First, we will investigate the approximation rate of B n,p (z) to the function ϕ in A 1 p -norm (Theorem 1), and by using the well-known Simonenko and Andrievskii method (see, for example, [6] and [18]), the approximation rate of B n,p (z) to the function ϕ in the uniform norm will be obtained (Theorems 2-6).

MATERIAL AND METHODS
Definition 1 [15, p. 97].The Jordan arc (or a curve) L is called a K-quasiconformal (K ≥ 1) arc (or curve) if there is a K-quasiconformal mapping f of a region D containing L such that f (L) is a line segment (or a circle).
Let F(L) denote the set of all sense-preserving plane homeomorphisms f of regions D ⊃ L such that f (L) is a line segment or circle and let where K ( f ) is the maximal dilatation of such a mapping f .Then L is K-quasiconformal if and only if gives the global definition of a K-quasiconformal arc or curve consequently.This definition is common in the literature.Through this work, the global definition will be considered.
Definition 2 [2].For given K 1, α 0, we say that G ∈ PQ(K, α) if L := ∂ G is expressed as a union of a finite number K j -quasiconformal arcs connected at the points z 0 , z 1 , z 2, ..., z m−1 , K = max 1≤ j≤ m K j , L is locally K-quasiconformal at z 0 , and in the local co-ordinate system (x, y) with origin at z j , 1 ≤ j ≤ m − 1, the following conditions are satisfied: It is clear from Definition 2 that each domain G ∈ PQ (K, α) may have m − 1 interior zero angles.If G does not have those interior zero angles (i.e α = 0), then G is bounded by a K-quasiconformal curve and we say that G ∈ Q (K) .Let G ⊂ C be a finite region bounded by Jordan curve L and let w = Φ(z) (w = ϕ (z)) be the conformal mapping of Ω := extG (G) onto The level curve (exterior or interior) can be defined for t > 0 as Let us denote Let L be a K-quasiconformal curve.By using the facts in ([15, p. 97]; [5, p. 76]; and [8, p. 26]) we can find a C(K)-quasiconformal reflection α(.) across L such that it satisfies the following Lemma 2 [1].Let L := ∂ G be a quasiconformal curve.Then for every z ∈ L there exists an arc β (z 0 , z) in G joining z 0 to z with the following properties: for every pair ξ 1 , ξ 2 ∈ β (z 0 , z).

APPROXIMATION IN THE A 1 p (G)-NORM
Suppose that G ∈ PQ (K, α) for some K 1 and α ≥ 0 is given.For the sake of simplicity, but without loss of generality, we assume that m = 2, z 1 = −1, z 2 = 1; (−1, 1) ⊂ G and let the local co-ordinate axes be parallel to Ox and Oy in the co-ordinate system Then z 0 is taken as an arbitrary point on L 2 (or on L 1 subject to the chosen direction).
We recall that the region G ∈ PQ (K, α) has interior zero angles in the nearest neighbourhood of each point z 1 = −1 and z 2 = 1, respectively.
We can say that the function w = ϕ(z) for the domain G ∈ PQ(K, α) satisfies the conditions described in Lemma 1 in the neighbourhood of point z 1,2 = ∓1.So, we can easily get On the other hand, using the properties of the function w = ϕ(z) in the neighbourhood of point z 1,2 = ∓1 (see [6,9]), we obtain Because each L j , j = 1, 2 is a K j -quasiconformal arc, α j (.) must be the quasiconformal reflection across L j .
Let us also denote: and where the constants a i , (i = 1, 2) are taken from Definition 2.
It is easy to check from Lemma 3 that for all ξ 1 , ξ 2 ∈ γ j i , i, j = 1, 2. Let N = N(R 0 ) be a sufficiently large natural number.For n > N and arbitrary 0 < ε < 1, let us choose R = r 0 + cn ε−1 such that r 0 < R < R 0 and points z j i , i, j = 1, 2 such that they are in the intersection of L R and γ j i .According to the positive direction on L R , these points divide L R into four parts as follows and γ j i (R) is a subarc of γ j i joining points ∓1 with z j i .Denote We can extend the function ϕ (z) to U in the following way Then, Then, using the above notations we obtain where Since the first part of (3.3) is analytic in G, there exists a polynomial P n−1 (z) where deg and let us take integrals over G of the p-th power of both sides From the Calderon-Zygmund inequality [5, p. 98], we obtain So, (3.5) and (3.6) give us Thus, (3.7) and (3.8) give We introduce the following notations: Theorem 1.Let G ∈ PQ (K, α) for some K, K 1 and α, 0 α < min 2(p−1) p+2 , 2 p+2 for p > 1.Then, for any number n = 2, 3, ... the p-Bieberbach polynomials B n,p (z) satisfy Proof.We shall obtain the proof by using (3.9).Firstly, assume that G ∈ PQ (K, α) for some K 1 and 0 ≤ α < min 2(p−1) p+2 , 2 p+2 .Let us choose g j (ξ ) := ϕ(ξ ) − ϕ((−1) j ), j = 1, 2, and ϑ (t) = t 1− 1+α p by using Lemma 2.4 [3], we obtain where i, j := mes γ j i (R).On the other hand, we have d( according to [17, Lemma 9.9].Then, from (2.1), (3.1), and (3.2) we get for all ε > 0. Combining (3.10) and (3.11), we have where i, j = 1, 2.
For sufficiently small 0 < ε 0 < ε 1 , we shall use the following notations: where Secondly, assume that G ∈ PQ(K; α) for some K 1, α ≥ 0. Then for all ε > 0 We get the following result by using these estimates and (3.9): let G ∈ PQ (K; α) for some K 1, 0 α < min 2(p−1)  p+2 , 2 p+2 .Then there exists a polynomial Q n (z) with Q n (0) = 0 for any number n = 2, 3, 4, ... such that where Now let us consider the polynomial From the Mean Value Theorem we have and by means of (3.12) we obtain So, if we consider the extremal property of the polynomials B n,p (z), then we have This gives the proof of Theorem 1.

RESULTS
For the given real numbers K 1, p > 1, and 0 α < 1 let us have the following notations: and Although the approximation rate in Theorem 2 and Corollary 1 is the same, p has a lower bound depending on arbitrary K 1 in Theorem 2 and K has an upper bound depending on arbitrary p, p ∈ 7  2 , 2 in Corollary 1.
Then the p-Bieberbach polynomials B n,p (z) (n 2) satisfy K and for some α, .

THE PROOF OF THEOREMS 2-6
To prove Theorems 2-6 we shall use a similar method to the one of Andrievskii and Simonenko employed in the proofs of analogous theorems for p = 2 (see [7,11] and [18]).

CONCLUSION
The main goal of this paper was to find the approximation rate of B n,p to ϕ in the uniform norm when the region has some certain singularity on the boundary.
It is seen from the theorems (Theorems 2-6) that the approximation rate depends not only on analytic properties (the quasiconformality coefficient K) but also on the geometric properties (the boundary has x α type zero angles) of the region.
Throughout this paper, c, c 1 , c 2 , ... are positive constants and ε, ε 1 , ε 2 , ... are sufficiently small positive numbers which in general depend on G.By the notation "a b" we mean that a ≤ c 1 b for a constant c 1 , which does not depend on a and b.The relation "a b" indicates that c 2 b ≤ a ≤ c 3 b, where c 2 , c 3 are independent of a and b.
.7) Let us consider two cases of p in the last double integral in (3.7) as: 1 < p < 2 and p ≥ 2. If 1 < p < 2, then using the Hölder inequality [21, p. 105] we obtain