Inverse Polynomial Images are always Sets of Minimal Logarithmic Capacity ∗ Klaus Schiefermayr†

Abstract In this paper, we prove that each inverse polynomial image (that is, each inverse image of an interval with respect to a polynomial mapping) is a set of minimal logarithmic capacity in a certain sense. Such sets play an important role in the theory of Pad´e-Approximation. The proofs are all based on the characterization theorems of Herbert Stahl.

Mathematics Subject Classification: 31A15, 30C10, 41A21 Keywords: Inverse polynomial image, Minimal logarithmic capacity, Symmetry property

1

Introduction

Let us formulate the following extremal problem. Problem (f, ∞): Let f be a complex function which is analytic in a neighborhood of ∞. Find an extremal domain D0 ⊆ C := C ∪ {∞} to which the function f can be extended in an analytic and single-valued manner. The domain D0 is extremal in the way that the complement K0 := C \ D0 has minimal logarithmic capacity. The paper is organized as follows. In Section 2, we introduce some notation and recall the main characterization theorem for problem (f, ∞) due to Herbert Stahl. In Section 3, we consider some important properties of inverse polynomial images and their relation to a certain quadratic equation (Abel-Pell equation). In Section 4, we prove a certain relation (called the symmetry property) for the Green function of the complement of an inverse polynomial image. Finally, in Section 5, we show that for each polynomial the corresponding inverse image is a set K0 of minimal logarithmic capacity with respect to a certain function f .

2

Stahl’s Characterization Theorem

In order to formulate for problem (f, ∞) the main characterization theorem due to H. Stahl, let us introduce some notation; see [11, Section 2.1]. Definition 1. Let f be a function which is analytic in a neighborhood of ∞. ∗

published in: Computational Methods and Function Theory 16 (2016), 375–386. University of Applied Sciences Upper Austria, School of Engineering, Stelzhamerstrasse 23, 4600 Wels, Austria, [email protected] †

1

Sets of minimal logarithmic capacity

2

(i) A domain D ⊆ C with ∞ ∈ D is called admissible for problem (f, ∞) if f has a single-valued analytic continuation throughout D. (ii) By D(f, ∞), we denote the set of all admissible domains D for problem (f, ∞). (iii) A domain D0 = D0 (f, ∞) ∈ D(f, ∞) is called extremal for problem (f, ∞) if cap(C \ D0 ) =

inf D∈D(f,∞)

cap(C \ D).

For any function f , which is analytic in a neighborhood of ∞, there exists a unique extremal domain D0 (f, ∞) ∈ D(f, ∞) with respect to problem (f, ∞). Both, existence and uniqueness, was proved by Stahl in the first and second part of [12], respectively, see also [11, Theorem 2]. The unique set K0 (f, ∞) := C \ D0 (f, ∞) is called the set of minimal logarithmic capacity with respect to problem (f, ∞). It was again Stahl [11, Theorem 11], who characterized the set K0 (f, ∞) in following way. Theorem 1. Let f be a function, which is analytic in a neighborhood of ∞, let D ∈ D(f, ∞) and K := C \ D. If K is of the form ∪ K = E0 ∪ E1 ∪ Jj j∈I

and satisfies the following five properties, then K = K0 (f, ∞), that is, K is the set of minimal logarithmic capacity with respect to problem (f, ∞). (i) For each z ∈ ∂E0 , there exists at least one analytic continuation of the function f out of D which has a non-polar singularity at z. (ii) For each z ∈ K \ E0 , there exist at least two analytic continuations of the function f out of D which lead to two non-identical function elements at z. (iii) All Jj , j ∈ I, are open, analytic Jordan arcs. (iv) The set E1 ⊆ K \ E0 is discrete in C \ E0 and each point z ∈ E1 is the endpoint of at least three Jordan arcs Jj . (v) For all z ∈ Jj , j ∈ I, ∂gK ∂gK (z) = (z), (1) ∂⃗n+ ∂⃗n− where gK (z) is the Green function for the complement of K, i.e. for D = C \ K, and with ∂⃗n∂+ and ∂⃗n∂− denoting the normal derivatives to both sides of the arc Jj . Remark 1. (i) Property (1) in Theorem 1 is usually called the symmetry property, see [11, Section 5.1]. (ii) Note that the minimal set K0 (f, ∞) plays a crucial role in the convergence theory of Pad´e-Approximants for the function f (z), see Stahl [13], Gonchar & Rakhmanov [2], Gonchar [1], Nuttall & Singh [4] and Nuttall [3]. √ Example. As an example, consider the function f (z) := 1/ z 2 − 1. Clearly, this function satisfies the assumption (analytic in a neighborhood of ∞), but can not be extended in an analytic and single valued manner to C \ {−1, 1}. Since f has the branch points −1 and 1, the set D(f, ∞) consists of all domains D ⊆ C such that ∞ ∈ D and such that the two points −1 and 1 are connected in the complement C \ D. It turns out that the set K0 of minimal logarithmic capacity is simply K0 = [−1, 1].

Sets of minimal logarithmic capacity

3

3

The Inverse Image and the Quadratic Equation of a Polynomial

Let us start with a lemma which states that any polynomial of degree n satisfies a certain quadratic equation, sometimes called the Abel-Pell equation. Since it is a direct consequence of the fundamental theorem of algebra, the proof is omitted. As usual, let Pn denote the set of all polynomials of degree n with complex coefficients. Lemma 1. For any polynomial Tn (z) = τ z n + . . . ∈ Pn , τ ∈ C \ {0}, there exists a unique ℓ ∈ {1, 2, . . . , n}, a unique monic polynomial H2ℓ (z) =

2ℓ ∏

(z − aj ) = z 2ℓ + . . . ∈ P2ℓ

(2)

j=1

with pairwise distinct zeros a1 , a2 , . . . , a2ℓ , and a unique polynomial Un−ℓ (z) = τ z n−ℓ + . . . ∈ Pn−ℓ such that the quadratic equation 2 Tn2 (z) − 1 = H2ℓ (z) Un−ℓ (z)

(3)

holds; the points a1 , a2 , . . . , a2ℓ are exactly those zeros of Tn2 −1 which have odd multiplicity. Further, there exists a monic polynomial Rℓ−1 (z) = z ℓ−1 + . . . ∈ Pℓ−1 such that Tn′ (z) = n Rℓ−1 (z) Un−ℓ (z).

(4)

Next, let us recall the notion of inverse polynomial images. For a polynomial Tn ∈ Pn , consider the inverse image of [−1, 1] defined by { } Tn−1 ([−1, 1]) := z ∈ C : Tn (z) ∈ [−1, 1] (5) and the inverse image of R defined by { } Tn−1 (R) := z ∈ C : Tn (z) ∈ R ,

(6)

respectively. In general, the inverse image Tn−1 (R) consists of n analytic Jordan arcs (running from ∞ to ∞) and m of these arcs are crossing in a point z0 if and only if z0 ∈ Tn−1 (R) and z0 is a zero of Tn′ with multiplicity m − 1. From this fact, it is easy −1 to conclude ∪ the following lemma, in which Tn ([−1, 1]) is decomposed into the sets E0 , E1 and Jj in such a way that Stahl’s theorem can be applied later on. For further properties of inverse polynomial images, especially concerning the connectivity, we refer to [8]. Lemma 2. Let Tn ∈ Pn and let H2ℓ be as in Lemma 1. Then Tn−1 ([−1, 1])

= E0 ∪ E1 ∪

ν ∪

Jj ,

(7)

j=1

where E0 , E1 , J1 , . . . , Jν have the following properties: (i) The set E0 is defined by { } { } E0 := z ∈ C : H2ℓ (z) = 0 = a1 , a2 , . . . , a2ℓ .

(8)

Sets of minimal logarithmic capacity

(ii) The set E1 is defined by where

(iii) The set

4

E1 := E1∗ ∪ E1∗∗ ,

(9)

} { E1∗ := z ∈ C : Tn (z) ∈ (−1, 1), Tn′ (z) = 0 , } { E1∗∗ := z ∈ C : Tn2 (z) − 1 = 0, Tn′ (z) = Tn′′ (z) = 0, H2ℓ (z) ̸= 0 .

(10)

∪ν

j=1 Jj

is defined by ν ∪

Jj := F1 ∪ F2

(11)

j=1

where

{ } F1 := z ∈ C : Tn (z) ∈ (−1, 1), Tn′ (z) ̸= 0 , { } F2 := z ∈ C : Tn2 (z) − 1 = 0, Tn′ (z) = 0, Tn′′ (z) ̸= 0 .

(12)

(iv) J1 , . . . , Jν are ν pairwise disjoint and open analytic Jordan arcs, and all endpoints of J1 , . . . , Jν are included in E0 ∪ E1 , more precisely: • If z0 ∈ E0 then z0 is a zero of Tn2 − 1 with multiplicity 2k − 1, k ∈ N, and exactly 2k − 1 of the ν arcs J1 , . . . , Jν have an endpoint in z0 . • If z0 ∈ E1∗ then z0 is a zero of Tn′ with multiplicity k, k ∈ N, and exactly 2k + 2 of the ν arcs J1 , . . . , Jν have an endpoint in z0 . • If z0 ∈ E1∗∗ then z0 is a zero of Tn2 − 1 with multiplicity 2k + 2, k ∈ N, and exactly 2k + 2 of the ν arcs J1 , . . . , Jν have an endpoint in z0 . Therefore, the number ν of analytic Jordan arcs is given by ν = (2ν1 + 2 + ν2 )/2 = ν1 + ν2 /2 + 1,

(13)

where ν1 is the number of zeros z0 of Tn′ (counted according to their multiplicity), for which Tn (z0 ) ∈ (−1, 1) and ν2 is the number of zeros of Tn2 − 1 (counted according to their multiplicity), which do not have multiplicity two. Note that ν2 is even. (v) For each j ∈ {1, . . . , ν} and each z0 ∈ Jj , z0 ∈ / F2 , there exists an ε > 0 such that Jj divides the disk Kε (z0 ) (center z0 , radius ε) into two parts, where in one part Im Tn (z) > 0 and in the other part Im Tn (z) < 0. Remark 2. (i) The set E0 is exactly the set of zeros of the polynomial H2ℓ , or, in other words, E0 is exactly the set of all zeros of Tn2 − 1 with odd multiplicity. (ii) Note that all points z0 ∈ E1∗ are zeros of the polynomial Rℓ−1 (z) given in Lemma 1. (iii) Note that each point z0 ∈ F2 ⊆ Tn−1 ([−1, 1]) is a simple zero of Tn′ , thus exactly two analytic Jordan arcs of Tn−1 (R) are crossing (at an angle of 90◦ ) in z0 . Since Tn (z0 ) ∈ {−1, 1}, on one analytic Jordan arc, z0 is a local minimum of Re{Tn (z)}, and on the other analytic Jordan arc, z0 is a local maximum of Re{Tn (z)}. Hence, z0 is a tie point for the combination of two open analytic Jordan arcs from F1 both lying on the same analytic Jordan arc of Tn−1 (R). In the following example, z0 = 0.0 is such a point.

Sets of minimal logarithmic capacity

5

Example. Consider the polynomial Tn (z) := 1 + z 2 (z − 1)3 (z − 2)4 of degree n = 9. Figure 1 shows the inverse images Tn−1 ([−1, 1]) (solid line) and Tn−1 (R) (dotted and solid line). • For z0 = 0.3009 . . ., we have Tn (z0 ) ∈ (−1, 1) and z0 is a zero of Tn′ with multiplicity k = 1; thus z0 ∈ E1∗ and 2k + 2 = 4 Jordan arcs have their endpoints in z0 . • The point z0 = 1.0 is a zero of Tn2 − 1 with multiplicity 2k − 1 = 3; thus z0 ∈ E0 and 2k − 1 = 3 Jordan arcs have their endpoints in z0 . • The point z0 = 2.0 is a zero of Tn2 − 1 with multiplicity 2k + 2 = 4; thus z0 ∈ E1∗∗ and 2k + 2 = 4 Jordan arcs have their endpoints in z0 . • The inverse image Tn−1 ([−1, 1]) is of the form (7), where (digits are cut off) { } E0 = −0.215, 1.0, 0.180 ± 0.421 i, 1.035 ± 0.787 i, 1.924 ± 0.729 i, 2.467 ± 0.301 i { } E1∗ = 0.3009 { } E1∗∗ = 2.0 and ν1 = 1, ν2 = 3 + 4 + 9 = 16 thus ν = ν1 + ν2 /2 + 1 = 10. The two Jordan arcs J1 and J2 on the real line are J1 = ]−0.215, 0.3009[

and

J2 = ]0.3009, 1.0[ .

Note that F2 = {0.0}.

Figure 1: Inverse images Tn−1 ([−1, 1]) (solid line) and Tn−1 (R) (dotted and solid line) for the polynomial Tn (z) := 1 + z 2 (z − 1)3 (z − 2)4

Sets of minimal logarithmic capacity

4

6

The Symmetry Property for Inverse Polynomial Images

Let us begin with an important remark, first concerning a certain complex square root and second about the Green function for the complement of an inverse polynomial image. For the definition of the Green function of a general subset of the complex plane and its crucial role in potential theory, we refer to [7]. √ Remark 3. (i) Let us consider the mapping z 7→ z 2 − 1. Here and in the following, √ let us define the complex square root z 2 − 1 so that it lies in the same quadrant as z (except for z ∈ [−1, 1], along which the plane must be cut). From this definition, we have √ √ 2 − 1 = − lim z z2 − 1 lim z→z z→z 0 Im(z)0

√ for each z0 ∈ [−1, 1]. Analogously, we define z → 7 Tn2 (z) − 1 for any polynomial √ 2 Tn ∈ Pn , thus Tn (z) − 1 depends continuously on z along every path that does not intersect an arc of Tn−1 ([−1, 1]). Consequently, for each z0 ∈ Tn−1 ([−1, 1]), we have √ √ Tn2 (z) − 1 = − z→z lim Tn2 (z) − 1. (14) lim z→z 0 Im Tn (z)>0

0 Im Tn (z) 0 for z ∈ C \ S. With the help of the above remark, we are able to prove the symmetry property for inverse polynomial images. Lemma 3. Let Tn ∈ Pn and define S := Tn−1 ([−1, 1]). Then the Green function gS for C \ S satisfies the symmetry property

for each z0 ∈ S \ (E0 ∪ E1 ) =

∪ν

∂gS ∂gS (z0 ) = (z0 ) + ∂⃗n ∂⃗n−

j=1 Jj ,

Proof. Define h(z) :=

1 n

(17)

where E0 , E1 , J1 , . . . , Jν are as in Lemma 2.

√ ( ) log Tn (z) + Tn2 (z) − 1

(18)

gS (z) = Re h(z).

(19)

then, by (15), Thus, for the derivatives, we have h′ (z) =

T ′ (z) 1 ·√ n n Tn2 (z) − 1

(20)

Sets of minimal logarithmic capacity

7

and, z = x + iy, ∂gS = Re h′ (z), ∂x (21) ∂gS = − Im h′ (z). ∂y ∪ν We consider the two cases (i) z0 ∈ j=1 Jj \ F2 and (ii) z0 ∈ F2 . ∪ (i) Let z0 ∈ νj=1 Jj \ F2 and let Jj ∗ be that unique Jordan arc which includes z0 . Let ( 1) ( ) ⃗n+ = nn12 be a unit vector normal to Jj ∗ in z0 and let ⃗n− = −n −n2 . Then

∂gS (z0 ) = ∂⃗n−

( ∂gS

∂x (z0 ∂gS ∂y (z0 ( ∂gS ∂x (z0 lim ∂g S ε→0+ ∂y (z0

∂gS (z0 ) = lim ∂⃗n+ ε→0+

) ( ) n1 , · n2 + ε(n1 + in2 )) ) ) ( − ε(n1 + in2 )) −n1 · . −n2 − ε(n1 + in2 )) + ε(n1 + in2 ))

By (7), (11), (12), Lemma 2 (v), (20), (21) and Remark 3, ( ∂gS lim

ε→0+

thus

∂x (z0 ∂gS ∂y (z0

) + ε(n1 + in2 )) + ε(n1 + in2 ))

( ∂gS = − lim

ε→0+

∂x (z0 ∂gS ∂y (z0

) − ε(n1 + in2 )) , − ε(n1 + in2 ))

∂gS ∂gS (z0 ) = (z0 ). + ∂⃗n ∂⃗n−

(ii) Let z0 ∈ F2 and let Jj ∗ be that unique Jordan arc which includes z0 . Then there exists a disk Kε (z0 ) (center z0 , radius ε) such that z0 is the only point which is a zero of Tn′ and of Tn2 − 1. The Jordan arc Jj ∗ divides Kε (z0 ) into two parts. In each part, h′ (z) given in (20) is continuous and limz→z0 h′ (z) exists. Thus, by (i), relation (17) follows.

5

Inverse Polynomial Images are always Sets of Minimal Logarithmic Capacity

In this section, we state and prove the main result of this paper. It says that the inverse image Tn−1 ([−1, 1]) of any polynomial Tn is the set of minimal logarithmic capacity with respect to a certain function f . Theorem 2. Let Tn ∈ Pn and let H2ℓ be as in Lemma 1. Then the extremal domain K0 (f, ∞) for the function 1 f (z) := √ (22) H2ℓ (z) is given by

K0 (f, ∞) = Tn−1 ([−1, 1]).

(23)

Sets of minimal logarithmic capacity

8

Proof. By the quadratic equation (3), the function f can be represented as Un−ℓ (z) f (z) = √ . Tn2 (z) − 1

(24)

∪ Let Tn−1 ([−1, 1]) = E0 ∪ E1 ∪ νj=1 Jj with E0 , E1 , J1 , . . . , Jν as in Lemma 2. Now, let us check the requirements (i)–(v) given in Theorem 1: { } (i): Since ∂E0 = E0 = a1 , a2 , . . . , a2ℓ , that is, ∂E0 is exactly the set of zeros of the polynomial H2ℓ , requirement (i) of Theorem 1 is satisfied. (ii): Now, let z0 ∈ Tn−1 ([−1, 1]) \ E0 . By (22), (24) and formula (14) of Remark 3, lim

z→z0 Im Tn (z)>0

f (z) = −

lim

z→z0 Im Tn (z)0

f (z) ̸= 0

and

lim

z→z0 Im Tn (z)>0

f (z) ̸= ∞.

(25)

The first / { relation of }(25) follows immediately from (22). By (22) and since z0 ∈ E0 = a1 , a2 , . . . , a2ℓ , the second relation of (25) follows. (iii): See Lemma 2. (iv): See Lemma 2. (v): See Lemma 3.

Remark 4. (i) Let a1 , a2 , . . . , a2ℓ ∈ C be 2ℓ pairwise distinct points in the complex plane and define 2ℓ ∏ H2ℓ (z) := (z − aj ). j=1

If there exist polynomials Tn ∈ Pn and Un−ℓ ∈ Pn−ℓ such that Tn2 (z) − 1 = 2 (z) holds then the set H2ℓ (z) Un−ℓ K0 := Tn−1 ([−1, 1]) has the property (∗) each point aj is connected in K0 with at least one other point ak . It is an immediate consequence of Theorem 2 that K0 has minimal logarithmic capacity among all sets with the property (∗). Moreover, note that, by Theorem 2 in conjunction with the uniqueness of K0 (f, ∞), the polynomials Tn and Un−ℓ , if they exist, are unique up to sign.

Sets of minimal logarithmic capacity

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(ii) Consider the case ℓ = 2 and let a1 , a2 , a3 , a4 ∈ C be the zeros (pairwise distinct) of H4 (z). The tuple (a1 , a2 , a3 , a4 ) ∈ C4 is called a Tn -tuple if and only if there exist 2 (z) holds. In polynomials Tn ∈ Pn and Un−2 ∈ Pn−2 such that Tn2 (z)−1 = H4 (z) Un−2 [10, Theorem 2], the author proved the following density result: Given four pairwise distinct points a1 , a2 , a3 , a4 ∈ C, there exist a ˜2 , a ˜3 ∈ C such that (a1 , a ˜2 , a ˜3 , a4 ) is a Tn -tuple and C1 C2 |a2 − a ˜2 | ≤ and |a3 − a ˜3 | ≤ n n hold for n ≥ N , where C1 , C2 , N depend only on a1 , a2 , a3 , a4 but do not depend on n. For the representation of Tn (z) and Un−ℓ (z) in terms of Jacobian elliptic and theta functions and other related results, see [6]. (iii) Closely connected to the problem mentioned in (i) is the following problem, sometimes called “(P´olya-)Chebotarev problem”: Given ν pairwise distinct points c1 , . . . , cν ∈ C in the complex plane, find a connected set S with c1 , . . . , cν ∈ S such that the logarithmic capacity cap S is minimal. If c1 , . . . , cν are exactly the simple zeros of Tn2 (z) − 1, where Tn is a polynomial of some degree n and if Tn−1 ([−1, 1]) is connected then S = Tn−1 ([−1, 1]). This was proved by the author in [9]. Finally, we consider the description of an inverse polynomial image (set of minimal logarithmic capacity) with the help of quadratic differentials, analogously as done by Stahl [11, Theorem 8]. Recall that a smooth arc Γ with parametrisation γ : [0, 1] → C, t 7→ γ(t), is a trajectory of the quadratic differential q(z) dz 2 if q(γ(t)) γ(t) ˙ < 0 for all t ∈ (0, 1). Theorem 3. Let Tn ∈ Pn and let Tn−1 ([−1, 1]) be as in Lemma 2. Then the Jordan arcs J1 , . . . , Jν of Tn−1 ([−1, 1]) are the trajectories of the quadratic differential q(z) dz 2 =

(Tn′ (z))2 1 · dz 2 . n2 Tn2 (z) − 1

(26)

Proof. By [11, Theorem 8], the corresponding quadratic differential can be represented by the derivative of the Green function, more precisely: ( ∂ )2 q(z) dz 2 = 2 ∂z gS (z) dz 2 , ∂ ∂ ∂ where ∂z = 12 ( ∂x − i ∂y ) is the usual complex (Wirtinger) differentiation. By (18) and ∂ (19), we have ∂z gS (z) = 12 h′ (z), thus, by (20),

(

)2 ( )2 (T ′ (z))2 1 ∂ , 2 ∂z gS (z) = h′ (z) = 2 · 2 n n Tn (z) − 1

which is the assertion. A direct proof (not using Stahl’s result) is the following: Let γ(t) = γ1 (t) + iγ2 (t), ˙ ̸= 0, t ∈ (0, 1). t ∈ (0, 1), be a parametrization of a Jordan arc Jj of Tn−1 ([−1, 1]) with γ(t) We have to prove that ]2 [ ′ ˙ Tn (γ(t)) γ(t) < 0, t ∈ (0, 1), Tn2 (γ(t)) − 1

Sets of minimal logarithmic capacity

10

holds. To this end, define { } u(x, y) := Re Tn (x + iy) , { } v(x, y) := Im Tn (x + iy) . Then ] [ ] ∂u (γ1 (t), γ2 (t)) · γ˙ 1 (t) + i γ˙ 2 (t) ∂x ∂y [ ] ∂u ∂u = γ˙ 1 (t) (γ1 (t), γ2 (t)) + γ˙ 2 (t) (γ1 (t), γ2 (t)) ∂x ∂y [ ] ∂u ∂u + i γ˙ 2 (t) (γ1 (t), γ2 (t)) − γ˙ 1 (t) (γ1 (t), γ2 (t)) . ∂x ∂y

Tn′ (γ(t))γ(t) ˙ =

[ ∂u

(γ1 (t), γ2 (t)) − i

Since v(γ1 (t), γ2 (t)) = 0, t ∈ (0, 1), we get, using the Cauchy-Riemann equations, } d{ v(γ1 (t), γ2 (t)) dt ∂v ∂v = γ˙ 1 (t) (γ1 (t), γ2 (t)) + γ˙ 2 (t) (γ1 (t), γ2 (t)) ∂x ∂y ∂u ∂u (γ1 (t), γ2 (t)). = −γ˙ 1 (t) (γ1 (t), γ2 (t)) + γ˙ 2 (t) ∂y ∂x { [ ]2 Thus Im Tn′ (γ(t)) γ(t)} ˙ = 0 and therefore Tn′ (γ(t)) γ(t) ˙ ≥ 0. Since γ(t) ˙ ̸= 0 and, by [ ′ ]2 ′ (11), Tn (γ(t)) ̸= 0, we get Tn (γ(t)) γ(t) ˙ > 0, which gives the assertion. 0=

Remark 5. Let Tn ∈ Pn and let H2ℓ , Un−ℓ , Rℓ−1 as in Lemma 1, then, by (3) and (4), 2 (z) R2ℓ−1 (z) 1 (Tn′ (z))2 1 n2 R2ℓ−1 (z) Un−ℓ . · = · = 2 (z) n2 Tn2 (z) − 1 n2 H2ℓ (z) H2ℓ (z) Un−ℓ

Acknowledgement. The author would like to thank the referee for pointing out an error in an earlier version of the paper and for several important suggestions.

References [1] A.A. Gonchar, Some recent convergence results on diagonal Pad´e approximants, Approximation theory, V (College Station, Tex., 1986), pp. 55–70. [2] A.A. Gonchar and E.A. Rakhmanov, Equilibrium distributions and the rate of rational approximation of analytic functions, Math. USSR-Sb. 62 (1989), 305–348. [3] J. Nuttall, Sets of minimal capacity, Pad´e approximants and the bubble problem, Bifurcation Phenomena in Mathematical Physics and Related Topics (C. Bardos & D. Bessis eds.), Reidel, Dodrecht, 1980, pp. 185–201. [4] J. Nuttall and S.R. Singh, Orthogonal polynomials and Pad´e approximants associated with a system of arcs, J. Approx. Theory 21 (1977), 1–42. [5] F. Peherstorfer, Minimal polynomials for compact sets of the complex plane, Constr. Approx. 12 (1996), 481–488.

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[6] F. Peherstorfer and K. Schiefermayr, Description of inverse polynomial images which consist of two Jordan arcs with the help of Jacobi’s elliptic functions, Comput. Methods Funct. Theory 4 (2004), 355–390. [7] T. Ransford, Potential Theory in the Complex Plane, Cambridge University Press, 1995. [8] K. Schiefermayr, Geometric properties of inverse polynomial images, Proceedings Approximation Theory XIII: San Antonio 2010, Springer Proceedings in Mathematics 13, 2012, pp. 277–287. [9]

, Chebotarev’s problem and inverse polynomial images, Acta Math. Hungar. 214 (2014), 80–94.

[10]

, A density result concerning inverse polynomial images, Proc. Amer. Math. Soc. 142 (2014), 539–545.

[11] H. Stahl, Sets of extremal capacity and extremal domains, http://arxiv.org/abs/1205.3811, 2012. [12]

, Extremal domains associated with an analytic function. I, II, Complex Variables Theory Appl. 4 (1985), 311–324, 325–338.

[13]

, The convergence of Pad´e approximants to functions with branch points, J. Approx. Theory 91 (1997), 139–204.