TR/07/90 August 1990 Asymptotic Behaviour of Zeros of Bieberbach Polynomials N. Papamichael, E.B. Saff and J. Gong

TR/07/90 August 1990 Asymptotic Behaviour of Zeros

of Bieberbach Polynomials

N. Papamichael, E.B. Saff and

J. Gong

Asymptotic Behaviour of Zeros of Bieberbach Polynomials

N. Papamichael * , E.B. Saff † ‡ and J. Gong †

Abstract LetΩ be a simply-connecteddomain in the complexplaneand letπ_n denotethen th degree Bieberbach polynomial approximationto the conformal map fofΩonto a disc. Inthis paper weinvestigatetheasymptoticbehaviour(as n →∞ )of thezerosof π_n,π_n' andalsoof thezeros of certain closely relatedrationalapproximantstof. Our results show that, ineach case,the distribution ofthezerosisgovernedbythelocationofthesingularitiesofthemappingfunction finC\ Ω,and wepresentnumericalexamplesillustratingthis.

Keywords : Bieberbach polynomials, Bergman kernel function, conformal mapping, zeros of polynomials.

* DepartmentofMathematicsandStatistics,BrunelUniversity,Uxbridge,MiddlesexUB8 3PH, U.K.

† nstituteforConstructiveMathematics,Departmentof Mathematics, University of South Florida, Tampa, Florida 33620, U.S.A.

‡ Theresearchofthisauthorwassupported in part by NSF grant DMS-881-4026 and by a Science and Engineering Research Council Visiting Fellowship at Brunel University.

BRUNELUNIVERSITY SEP 1991

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1 Introduction

Let Ω beasimply-connecteddomainofthecomplexplaneC,whoseboundary∂Ω isa closed Jordan curve, and letζ∈Ω.Then,bytheRiemannmapping theorem,there exists a unique conformal mapping w=f_ζ (z) of onto a disc { w: |w| <Ω , }, such that r_ζ

=0, =1.

) ζ

ζ(

f f_ζ'(ζ)

The radius of this disc is called the conformal radius of r_ζ Ω with respect toζ. For the inner product

, ) ( ) ( :

) ,

(g h =∫∫Ωg z h z dm

Where dm is the 2-dimensional Lebesgue measure, we consider the Hilbert space L²(Ω):={g : g analytic in Ω , || g ||² = ( g , g ) < ∞ } .

Let K( z , ) denote the Bergman kernel function of ζ Ω which has the reproducing property g( )ζ = ( g, k (·, )), ζ ∀g∈L²(Ω). (1.1) (cf. [1], [3], [4], [8]). Then it is known (cf. [4, p.34]) that = (r_ζ πΚ(ζ,ζ))^−1/2and that for z ∈Ω

∫=

=

= ^z

t K t dt

z K K f

z z K

fζ ζ ζ (,ζ) .

) ζ , ζ ( ) 1 ( ), ζ , ζ (

) ζ , ) (

'(

(1.2)

Next letQ_n(z) =γ_n zⁿ +...,γ_n>0, be the sequence of orthonormal polynomials for the inner product ( · , · ) , ie.

. )

( )

( _{1 k,l}

k z Q z dm

Q =δ

∫∫^Ω

Since is a Jordan region, it is known ( cf. [4, p. 17 ] ) that Ω {Q_n}^∞₀ forms a complete orthonormal systemforL²(Ω)andconsequently, from the reproducing property (1.1),thatK(.,ζ) has the L²(Ω)- convergent Fourier series expansion

∑^∞

=

−

0

. ) ( ) ζ ( )

ζ , (

j

j

j Q z

Q z

K (1.3)

TheBieberbach polynomialsπ_n forΩare defined by , ) ζ , ) (

ζ , ζ ( : 1 ) (

2 1 1

dt t K K

z

t n n

n

∫

=

−

−

=

ζ

π (1.4a)

WhereK_n−1(,.ζ)denotes the partial Fourier sum

).

( } ζ ( :

) ζ ,

( ¹

0

1 z Q Q z

K _j

n

j j

n ∑⁻

=

− = (1.4b)

These polynomials satisfy

), ζ , ζ (

|| 1

||

1 ) ζ ( , 0 ) ζ (

1 2 ' '

−

=

=

=

n n

n n

π K π

π

andprovideapproximations tothemappingfunction f_ζ in the sense thatπ_n → f_ζ locally uniformly in Ω ( cf. [4, p.34 ] ). The latter is a directconsequence ofthefactthat convergencein the norm of L²(Ω) implies uniform convergence on each compact subset of Ω (cf. [4, p. 26 ]). More generally, if

∞

}1

{sj is anycomplete orthonormal system for L²(Ω) and

∫

−

−

= ²

ζ 1 1

, ) ζ , ˆ ( ) ζ ζ, ˆ ( : 1 ) ˆ (

t n n

n K t dt

z K

π (1.5a)

Where

, ) ( ) ζ ( )

ζ ,

ˆ ( ¹

1

1 z s s z

K _j

n

j j

n ∑⁻

=

− = (1.5b)

then πˆ_n→ f_ζ locally uniformly in Ω (cf. [4, p.32 ] ).

In practicethesuccessof theabovemethodfor approximating f_ζ depends critically on the choice of the denning orthonormal system. In particular, if the mapping function f_ζ has singularities eitherontheboundary∂Ω orcloseto∂Ω in C \ Ω, then it is essential that the orthonormal system containsfunctionsthatreflectthecorresponding singularities of K(,.ζ) (cf. [7], [9], [10]). For this reason, the problemofdeterminingthelocationandnatureof thesingularities of f_ζ is of considerable practical interest.

The purposeof thispaper is todescribe theasymptoticbehaviour (asn→ ∞) of the zeros of theBieberbach polynomials π_n,, and also of the zeros of certain rational approximantsπ_n,of the type studied in [7], [9] and [10] ( see also [4, p.36 ] ). Our results show that the distributions of these zeros, and also of the zeros of the derivatives π_n^' and πˆ_n^',are governed by the location of the singularities of the mapping function f_ξ in C \ Ω. From the more practical point of view, the results of several numerical experiments indicate that the distributions of the zeros of π_n andπ_n' can help to determine the approximate location of these singularities.

The paper is organized as follows: Section 2 contains the statements of our main result concern- ing the zeros of Bieberbach polynomials (Theorem 2.2 ) and of an intermediate result (Theorem 2.1) which is needed for our proofs. In Section 3 we present three examples illustrating the results stated in Section 2,and makeseveral observations regarding the distributions of the zeros of π_nand π_n'in rela- tion to the singularities of f_ξ . In Section 4 we give the proofs of Theorems 2.1 and 2.2. Finally, in Section 5 we treat the problem of the distribution of zeros of rational approximants πˆ of the type stu- _n died in [7], [9], and [10].

Our main results will be given in terms of a normalized counting measure for zeros, which is defined as follows: If P is a polynomial of degree n with zeros z1, z2, . . . . z_n ( some of which may be repeated), then the measure v(P) is defined by

v(P)(B) :=^1.

n [ # of zeros of P in B ] , (1.6) for any Borel set B ⊆ C. Thus, v(P) is a probability measure on the Borel subsets of C.

-3-

2 Statements of Results for Bieberbach Polynomials

Let w = ф(z) denote the conformal mapping of D := C\Ωonto { w : | w | > 1 }, normalized by ф(∞)=∞ and φ′(∞)>0, and observe that the Green function of D with pole at is given by ∞

.

| (z)

| log ) ,

(z ∞ = φ

g_D Further, for each σ > 1, let Гσ denote genetically the locus },

log ) , ( g : {z }

| (z) :|

{

: σ σ _D σ

σ = = = ∞ =

Γ z z (2.1)

and set Γ₁:=∂Ω. Finally, let Ω_σ denote the collection of points interior to the level curve Γ_σ. For the compact set Ω_σ,σ>1,there exists a unique probability measureµ_σ supported on Γ_σ that minimizes the energy integral

∫∫

= log| | ( ) ( ) :

]

[ z t ¹ d z d t

I µ µ µ (2.2)

over all probability measures supported on Ω_σ ( cf. [5, § 16.4 ], [13] ). The measureµ_σis called the equilibrium distribution for Ω_σ and the logarithmic capacity of Ω_σ is definned by

]).

µ [ exp(

: )

(Ω^___σ = −Ι _σ

cap (2.3)

In terms of the mapping functionΦ we have that

cap(Ω_σ)=σ /Φ^'(∞). (2.4)

Before giving our main result it is convenient to state

Theorem 2.1 With the notations and assumptions of Section I, ),

1 ρ(

| 1 ) ζ (

| sup

lim ^1/ = <

∞

→

n n n

Q (2.5)

where ρ(>1)) is the largest index such that f_ζ has an analytic ( single-valued) extension throughout

ρ. Ω

Notice that iff_ζ has a singularity on Γ₁ =∂Ω,, then ρ= 1. Moreover, if ρ< ∞, thenf_ζ has at least one singularity on Γ_p.

We can now state our main result concerning the zeros of Bieberbach polynomials.

Theorem 2.2 Suppose that the constant ρ of Theorem 2.1 is finite and let A ⊆ N be a sequence for which

1 .

| ) (

|

lim ^1/

ζ = ρ

∈∞

→

n n A n

n Q (2.6)

Then in the weak-star topology of measures, the normalizing counting measures for the zeros of π_n and π 'n satisfy

( )₋1 →∗ and v

( )

vπ_n µ_ρ π_n µ_ρ (2.7)

Where is the equilibrium distribution for µ_ρ Ω_p. In (2.7), the first convergence means that

∫

∫

∈→∞ f dv π_n f dµρ A

n

n ( )

lim ₁

for every function f continuous onChaving compact support. From this it follows (cf. [6, pp. 8,9] ) that if B is any Borel set, then

), ( µ ) )(

π ( sup lim ) )(

π ( inf lim ) (

µ_ρ B⁰ v ₁ B v _{n 1} B _ρ B

n n

n

<

<

< ₊

∞ + →

∞

→ o

B denotes the interior of B.

where

Since supp (µ_p)=Γ_ρ,, we immediately deduce from (2.7) the following.

Corollary 2.3 With p as in Theorem 2.2, every point on Γ_ρ is an accumulation point of the zeros of theBieberbachpolynomials{π_n}^∞₀ and of the derived sequence {π^'n}₁^∞ Consequently, {π_n}∞₀ can- not converge uniformly in any neighbourhood containing a point of Γ_ρ.

Remark 1 Theorem 2.2 (and Corollary 2.3 ) does not preclude the possibility that there exist accu- mulation points of zeros of {π_n} or {π^'_n} that lie off of the level curve Γ_ρ. However, the number of zeros of {π_n+1}_n∈Λ and of{π^'_n+1}_n∈Λ that can lie on a given compact set disjoint from Γ_ρ is o (n).

Remark 2 For p > 1, Corollary 2.3 also follows from the maximal geometric convergence of the sequences {π_n}^∞₀ and {π^'_n}₁^∞and Walsh's extension of the Jentzsch theorem [15]. However, this argument does not apply to the important case p = 1.

3 Examples

3.1 Consider the case where = { z : | z | < l } . Then Ω ,..., 1 , 0 π ,

) 1

( = n+ z n=

z

Q_n ⁿ

and hence

, , ) ζ ζ 1 ( . 1 π ) 1 ζ ( ) 1 ( 0 π. ) 1 ζ ,

( ₂ ∈Ω

= − +

=

∞

=

∑

j z

k ^j

2 1 1

0

1 (1 ζ )

1 ) ζ )(

1 ( ) ζ . ( π ) 1 ζ )(

1 ( π. ) 1 ζ ,

( z

z n z

z n j

z k

n n

j n

j

n −

+ +

= − +

= ^{− +}

=

−

∑

Also,

- 5 - ζ , 1

| ζ ζ

| 1 ( )

( ²

ζ ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛

−

− −

= z

z z f

so that the mapping function f_ζ has a simple pole at the point z = 1/ζ but is otherwise analytic in the extended plane. Therefore:

(i) Since Φ(z) = z, the constant p in Theorems 2.1 and 2.2 is

|. ζ

|

= 1 p

This is, indeed, the reciprocal of

.

| ζ

|

| π ζ

| 1 lim

| ) ζ (

|

lim ^1/ = + ^1/ =

→∞

→∞

n n n

n n n

Q n

(ii) If ζ ≠ 0, then according to Theorem 2.2 ( which holds with Λ= N ) , µ ) ' ( µ

)

(π_n →^* _1/ζ and v π _n →^* _1/ζ v

where µ_1|/ζ| is the equilibrium distribution for the disc |z|<1/|ζ|. That is,

π , 2

| ζ µ_1/|ζ| | ds

d =

where s denotes arc length on the circle |z|=1/|ζ|. In other words, dµ_1|/ζ| is the uniform distribu- tion on the circleζ This limit behaviour can be verified directly from the explicit formulae forπ^'_n(z)={k_n−1(z,ζ)/k_n−1(ζ,ζ)}and π_n(z).

3.2 Let Ω be the rectangleΩ = { z =x + iy ; \ x \ < 2 , \ y \ < 1 } and set ζ= 0. Then, the mapping function f0 is analytic on∂Ω, but its analytic extension has a simple pole at each of the points

z = 2 ( 2 k +i l) , k , 1 = 0 , ± 1 , ± 2 , . . . , k + l= o d d .

Thus,the singularities of f0nearest to∂Ωoccur at the pointsz = ±2i and, consequently, the value of ρ in Theorems 2.1 and 2.2 is (to 5 significant digits )

4095 . 1 ) 2 ( ρ=Φ i = (cf.[10,p.662]).

In Figure 1 we have plotted the zeros of the Bieberbach polynomialsπ₁₇andπ₂₉and in Figure 2 those of their derivatives π'₁₇ and π'₂₉. These zeros were obtained by using the Fortran conformal mapping package BKMPACK of Warby [16] ( for computing the Bieberbach polynomials ) and the NAG zero finding subroutine C02AEF.

Figures 3 and 4 contain, respectively, plots of the images of the zeros of π₁₇,π₂₉ (with the exception of z = 0 ) and of π'₁₇, π'₂₉,under the conformal map ф: C \ Ω→{w |:w|>1}. These images were obtained from an accurate approximation to ф, which was sagain computed by using BKMPACK

(a) n =17 (b) n=29

Figure 1: Zeros of π_n

(a) n = 17 (b) n = 29

Figure 2 : Zeros of π^'_n

-7-

(a) n=17 (b) n=29

Figure 3 : Images of zeros of π_n

(a) n=17 (b) n =29

Figure 4 : Images of zeros of π_n^'

We observe the following in connection with the plots in Figures 1-4 :

(i) The plots in Figures 3 and 4 illustrate the positionof the images of thezerosrelative to the circles | w| = 1 and | w | = P = 1.4095, and hence the closeness of the zeros to the level curve Γ_ρ.

(ii) As predicted by Theorem 2.2, the zeros of the Bieberbach polynomials appear to be approaching the level curve that corresponds to the nearest singularities of fΓ_ρ 0. Although the zeros appear to thin out near the two singular points ±2i, where f0 becomes unbounded, Theorem 2.2 assures us that they do approach these points as n increases.

(iii) The behaviour ofthezerosof the derivatives is similar to that described above, except that now there is always a zero close to each of the four corners of Ω . This reflects the fact that f0' is zero at each of these points.

3.3 LetΩbe the L-shaped domain illustrated in Figure 5 and take ζ =0. In this case, the map- ping function f0 has a branch point singularity at the re-entrant corner z_c = 1 , in the sense that

. ,

) ( ) ( )

( ₀ ^2/3

0 z f zc z zc as z zc

f − − − →

In addition, f0has simple pole singularities in C\Ω, of which theclosest to ∂Ω occur at the points ]).

p.663 [10, . cf ( 1±l

− Since f0 has a singularity on Γ₁ =∂Ω, it follows that ρ =1 in Theorems 2.1 and 2.2.

Figure 5 : L-shaped region Ω

In Figure 6 we have plotted the zeros of the Bieberbach polynomials π₁₃ and π₂₃ and in Figure 7 those of their derivativesπ'₁₃ andπ'₂₃. These zeros were again computed by using the conformal mapping package BKMPACK and the NAG zero finding routine C02AEF .

-9-

(a) n = 13 (b) n = 23

Figure 6 : Zeros of π_n

(a) n = 13 (b) n = 23

Figure 7: Zeros of π_n^'

AspredictedbyTheorem2.2,thezerosof boththeBieberbachpolynomials and their derivea- tives appeartobeapproachingtheboundary ∂Ω.In bothcases,thezerosappeartothinoutnear the re-entrantcorner zc = 1, where f0' becomes unbounded. They do, however, approach zc as n increases.Asimilar,butlesspronounced,thinningoutoccursnearthepartsof ∂Ω whichareclose to thetwopoints-1±i ,where f0becomesunbounded.Finally,inthecaseofthederivatives,there are always zeros close to each of the right-angled corners, reflecting the fact that f0' is zero there.

4 Proofs of Theorems 2.1 and 2.2

To establish Theorems 2.1 and 2.2 we shall make use of several lemmas. The first is due to S.N. Bernstein and J.L. Walsh.

Lemma 4.1 ( [14, p.77 ] ) Let E ⊂ C be a compact set whose complement U := C\E is connected andregularwith respectto theDirichletproblem. Let g_U(z ,∞) denote the Green function for U with pole at ∞. Pn is a polynomial of degree at most n and

,

| ) ( max |

|| :

||P_{n L∞(E)} =Z ∈E P_n z <M then

. )},

, ( exp{

)

(z M ng z z U

P_n < _U ∞ ∈ (4.1)

Lemma 4.2 ( [14, p.28 ] ) If G is a bounded simply-connected domain and R > 1 is given, then there exists a closed Jordan region E ⊂G such that the closed regionG lies interior to the level curve

}.

log ) , ( : {

: z g _\ z R

l_R = _CE ∞ = (4.2)

Combining the above lemmas we shall establish

Lemma 4.3 The orthonormal polynomials Qn of Section 1 satisfy

lim|| ||^1/ 1.

)

( =

∞

→ ^{−∞ Ω}

n n L

Q

n (4.3)

ProofTheargumentissimilar to that in [14, p.96 ]. Let R > 1 be given. Then, by Lemma 4.2, there existsaclosedJordanregionE ⊂Ω such that Ω lies interior to the level curve lR of

(4.2).Let r:= dist (E , ∂Ω). From the well-known estimate

1 | | , ,

| ) (

| ² ₂ g ² dm z E

z r

g <π ∫∫Ω ∈ ^(4.4)

which holds for everyg∈L²(Ω)(cf.[4,p.4]),we get

-11-

π ,

| 1 π |

|| 1

|| ² _{( ) 2} ² ₂

dm r r Q

Q_{n L}⁻ _E < ∫∫^Ω _n = and so, by Lemma 4.1,

π.

||

|| _{( )}

r Q R

n l

L

n _{∞ R} < (4.5)

Since Ω lies interior to lR, we have by the maximum principle that .

||

||

||

|| _{( )}

) n L l^R

L

n Q

Q _−∞Ω < _∞ Thus, from (4.5),

.

||

||

sup

lim Q_n ¹_L^/n_{( )} R

n

Ω <

∞

→ ^∞

But as R is arbitrary, letting R↓1 yields

. 1

||

||

sup

lim ^1/_(Ω) <

∞

→ ^∞

n n L n

Q Moreover, the inequality

1

||

||

inf

lim ^1/_(Ω) >

∞

→ ^∞

n n L

n Q

is an easy consequence of the fact that the Qn ’s are orthonormal. Thus, (4.3) holds.

□

In the terminology of [11, §3 ], Lemma 4.3 shows that the measure dm on Ω is completely regu- lar. Consequently ( cf. [11, Proposition 3.2 ] ), the leading coefficients of the polynomials Qγ_n n

satisfy

). ( γ 1

lim ^1/

= Ω

∞

→ cap

n

n n (4.6)

Weremarkthatinthespecialcase when Ω is bounded by an analytic Jordan curve, then estimates finer than that in (4.6) can be obtained for the 's (cf. [4, p.12 ] ).γ_n

We can now give the :

Proof of Theorem 2.1 Again the proof is essentially the same as that given by Walsh [14, p.130]. From (1.2) and (1.3), we have

).

( ) ζ ( )

( ) ζ , ζ (

0 '

ζ z Q Q z

f

K _{n n}

n

∞

∑

− (4.7)

) (ζ

Qn are the Fourier coefficients of the functionK(ζ,ζ)f_ζ '(z):

ie. the constants

∫∫Ω

= (ζ,ζ) . )

ζ

( K f_ζ^'Q dm

Q_{n n} (4.8)

Nowsupposethatp>1sothat f_ζ(and hence ) is analytic on f_ζ^' Ω_ρ ⊃Ω, and let pn denote the poly- nomials of respective degrees at most n of best uniformapproximationto f_ζ^' onΩ,. From a result of Walsh [14, p.90 ], we have

1 .

||

||

sup

lim ^' ₁ ^1/_{( )}

ζ − _{− Ω} < ρ

∞

→ ^∞

n n L n

p

f (4.9)

Furthermore, by the orthogonality property of the Qn' s, equation (4.8) can be written as dm

Q p f K

Q_n(ζ)= (ζ,ζ)∫∫_Ω( _ζ^'− _n−1) _n (4.10) Thus, from (4.10) and the Cauchy-Schwarz inequality, we get

lim sup .

ρ ) 1 ζ ( ^1/n ≤

Qn (4.11)

Notethat in the case when f_ζ is not analytic on Ω, that is = 1, inequality (4.11) remains valid. ρ Next, we suppose that

ρ. 1 ) 1

ζ ( sup

lim ^1/ = ≤

→∞ σ

n n n

Q (4.12)

andshowthat this leads to a contradiction.Indeed,from (4.3) and Lemma 4.1, we have for σ >τ > ρ ,

sup

lim ^1/_{( )} τ

τ ≤

∞ Ω

→ ^∞

n n L n

Q and so, by (4.12),

Thus,the seriesin(4.7)convergesuniformly on Ω_τ toan analyticextensionof K(ζ,ζ)f_ζ^'But this contradicts the definition of as being the largest index forρ which f_ζ (and, equivalently,) is ana- lytic throughout Ωρ

□

. 1 ).

( ) ζ ( sup

lim ^1/_(Ω)≤ <

→∞ ^∞ σ

τ

τ

n n L n n

Q Q

In the proof of Theorem 2.2 we shall make use of the following result due to Blatt, Saff and Simkani, which generalizes an earlier theorem of G. Szeg [12]. ¨o

Lemma4.4([2])LetSbeacompactset with positive capacity and suppose that the monic polyno- mials Pn (z) = z + … , which are given for a subsequence of indices n, sayⁿ n∈∧ ⊆ N, satisfy (a) limsup ^1/_{( )}≤ ( ), ∈∧,

∞ ∞

→ P_{n L}ⁿ_S cap S n

n

and

(b) limv( )(A)=0, ∈∧,

∞

→ P_n n

n forallclosedsets Acontainedinthe (2-dimensional) interior of the polynomial convex hull of the set S.

Then, in the weak-star sense,

v(P_n)→^* µ_s, as n→∞, n∈∧,

where µ_s is the equilibrium distribution for S.

By the polynomial convex hull of S we mean the complement of the unbounded component of .

\S C