#### Abstract

The Ahlfors map is a conformal mapping function that maps a multiply connected region onto a unit disk. It can be written in terms of the Szegö kernel and the Garabedian kernel. In general, a zero of the Ahlfors map can be freely prescribed in a multiply connected region. The remaining zeros are the zeros of the Szegö kernel. For an annulus region, it is known that the second zero of the Ahlfors map can be computed analytically based on the series representation of the Szegö kernel. This paper presents another analytical method for finding the second zero of the Ahlfors map for an annulus region without using the series approach but using a boundary integral equation and knowledge of intersection points.

#### 1. Introduction

A conformal mapping that maps a multiply connected region of connectivity onto a unit disk is known as the Ahlfors map. It generalizes the Riemann map for a simply connected region. The Ahlfors map with a base point is a -to-one map. It maps each boundary of corresponding in a one-to-one manner onto the boundary of the unit disk and maps to the origin . The Ahlfors map of a multiply connected region has several applications in modern function theory. For examples, the Bergman kernel and Green’s function of a multiply connected region can be written in terms of finitely many Ahlfors map [4, 5]. The Ahlfors map has zeros in where one of the zeros is which can be freely prescribed, and the remaining zeros in are unknown . It is known that the Ahlfors map can be written in terms of the Szegö kernel and the Garabedian kernel. In , the zeros of the Ahlfors map with a base point occur at the pole of the Garabedian kernel at and the remaining zeros are those of the Szegö kernel.

Kerzman and Stein  and Kerzman and Trummer  have shown that the Szegö kernel of a simply connected region satisfies an integral equation which is valid also for a multiply connected region . The integral equations with the Kerzman–Stein kernel, Neumann kernel, and Szegö kernel related to the Ahlfors map for a doubly connected region have been derived in . The integral equation with the generalized Neumann kernel for computing the Ahlfors map has been constructed in . The method in  is useful provided the zeros are known. Several integral equations involving the Kerzman–Stein kernel, the Neumann kernel, the Szegö kernel, the Neumann-type kernel, and the Kerzman-Stein type kernel have also been derived in [11, 12].

In , the formula of the second zero of the Ahlfors map for a doubly connected region has been derived in terms of an integral involving the Szegö kernel and the derivative of the Szegö kernel, but the second zero has not been computed numerically. In , the second zero of the Ahlfors map for an annulus region has been obtained analytically from the series representation of the Szegö kernel. The problem of finding the zeros of the Ahlfors map has not been studied in , but discussed in . The second zero of the Ahlfors map for any doubly connected region has been computed numerically in  based on the integral equation and Newton iterative method.

In this paper, we derive a boundary integral equation related to the Ahlfors map for a doubly connected region different from . This integral equation together with knowledge of intersection points provides another analytical method for finding the second zero of the Ahlfors map for an annulus region. Unlike , the method does not depend on the series representation of the Szegö kernel.

The organization of this paper is as follows: Section 2 presents some auxiliary materials related to the Ahlfors map and the zeros of the Ahlfors map. In Section 3, we give a boundary integral equation related to the Ahlfors map for a doubly connected region. In Section 4, we present another analytical method for finding the second zero of the Ahlfors map for an annulus region. In Section 5, we give a conclusion.

#### 2. Auxiliary Materials

Let be a bounded doubly connected region with the boundary consisting of two smoothly closed Jordan curves i.e., The curve lies in the interior of the boundary The outer curve has a counterclockwise orientation, and the inner curve has a clockwise orientation. The curves are parameterized by twice continuously differentiable complex-valued functions with the first derivatives The total parameter domain is defined as the disjoint union of two intervals of The parameterization of the whole boundary on is defined as The unit tangent to the boundary at is given by

It is known that the Szegö kernel satisfies the Kerzman–Stein integral equation [1, 6]:where

The function is continuous on the smooth boundary of , and it is known as a Kerzman–Stein kernel [6, 7]. If and are on a circle, then Numerical implementations of computing the Szegö kernel based on (2) are discussed in . With and (2) becomes

The derivative of the Szegö kernel has been derived in  as a solution of the integral equation:where

Let be the Ahlfors function which maps conformally onto the unit disk which satisfies the conditions , and where are the zeros of the Ahlfors map and can be freely chosen. The boundary values of is represented bywhere , are the boundary correspondence functions of the Ahlfors map on

Differentiating both sides of (7), we have

Taking modulus on both sides of (8) gives

Dividing (8) by (9) and using (1), we get

The image of remains in a counterclockwise orientation, so , and the image of is in reversed orientation, so

Since thus from (10), we get

In general,

The interior values of the Ahlfors map which is analytic on can be obtained by using the Cauchy integral formula:

For points closed to the method of Helsing and Ojala  can be used for the efficient numerical computation of (13).

The Ahlfors map can be represented in terms of the Szegö kernel and the Garabedian kernel It is given as follows :

Since , (14) can be written as

It is shown that the boundary values of the Ahlfors map can be determined from the boundary values of the Szegö kernel since the boundary values of the Garabedian kernel can be determined from the boundary values of the Szegö kernel.

Differentiating both sides of (15), we get

In , an alternative formula for the derivative of the boundary correspondence function of the Ahlfors map has been derived by using (8), (15), and (16), i.e.,

In general, the zeros of the Ahlfors map are unknown except for an annulus region. In , the second zero of the Ahlfors map for a doubly connected region is represented by

However, numerical computation of based on (18) is not given in  where the boundary values of can be computed by solving the integral equation (5).

The second zero of the Ahlfors map for an annulus region is shown to be , and it has been obtained from the series representation of the Szegö kernel :

Another series representation of the Szegö kernel for an annulus region has been derived in  by solving the Kerzman–Stein integral equation (2) using the Adomian decomposition method, and it is given by

The two series (19) and (20) are shown to be equivalent. However, the series in (20) gives a faster convergence compared to the series in (19) .

In this section, a new boundary integral equation related to the Ahlfors map for a doubly connected region different from  is derived and applied in Section 4 to determine analytically the second zero of the Ahlfors map. It is given in the following theorem.

Theorem 1. For all the function of the Ahlfors map for the doubly connected region satisfieswhere and are the zeros of the Ahlfors map.

Proof. The Ahlfors map for a doubly connected region can be written aswhere is analytic in and Applying log on both sides of (22) givesDifferentiating on both sides of (23), we getObserve thatApplying the residue theory , Sokhotskyi formula , and (24) to (25), we getLetting and and multiplying (26) by givesSince from (8), thus, (27) yieldsThis completes the proof of Theorem 1.

If we take the imaginary part on both sides of (21), then it reduces the integral equation derived in , i.e.,whereand is defined in (17). The kernel is the classical Neumann kernel. If is a simply connected region, then (29) reduces to the Warschawski’s integral equation for the Riemann map as given in  (p. 394-395) with i.e.,

Next we show how to find the second zero of the Ahlfors map for an annulus region using (21) and knowledge of intersection points.

#### 4. Finding the Second Zero of the Ahlfors Map for an Annulus Region

We represent the left-hand side and the right-hand side of (21), respectively, as

We consider an annulus region bounded bywhere and Given is known, and we assume is unknown. In the next two examples, we show the graphs of for special values of and The graph of depends on which is computed based on (17). The function in (17) depends on and which are computed by solving (2) and (5), respectively. Thus, the graph of does not depend on If is known, then the graph of is the same as the graph of

Example 1. Given and and the annulus region is shown in Figure 1. The graphs of the functions and are shown in Figure 2.

Example 2. Given and and the annulus region is shown in Figure 3. The graphs of the functions and are shown in Figure 4.
Figures 2 and 4 show some common characteristics:(i)C1: the graphs of and have the same shape and size(ii)C2: the graphs of and are symmetrical with respect to the imaginary axis(iii)C3: both graphs also have intersection points on the imaginary axis(iv)C4: the graph of is 4i units above the graph of These characteristics are unique for the Ahlfors map for the annulus region. Note that the graphs in Figures 2 and 4 can also be generated using (33) provided is known. In Example 3, we consider plotting (33) using

Example 3. Given and and the annulus region is shown in Figure 3. We choose The graphs of the functions and are shown in Figure 5.
In Example 4, we consider plotting (32) involving nonconcentric circles.

Example 4. Consider the region bounded by nonconcentric circles and with and The nonconcentric circles region is shown in Figure 6. The graphs of the functions and are shown in Figure 7.
The graphs in Figures 5 and 7 no longer satisfy the characteristics C1, C2, C3, and C4. Based on these observations, we prove the following theorem to determine the second zero of the Ahlfors map for an annulus region.

Theorem 2. Let be defined in (33), where , and be given. Then, there exists and such thatFurthermore,

Proof. From (33),We consider for two cases which are , and

Case 1.
Let , and be given, and is unknown. Since is periodic, there exists with such thatThis givesFrom (38) and (39), respectively, we haveFrom (40) and (41), we seek and such thatthat is,Since thus this impliesChoosing givesThen,Using (42) and (46) into (40) and (41) givesSimplifying (48) to solve we getSince and we haveSolving (50) and applying (45) givesChoosing givesThen, for we haveApplying (46), (50), (53), and (54) into (47), we getIt shows that This impliesor

Case 2.
Let If is periodic, then there exists , such thatThis givesFrom (59) and (60), respectively, we haveFrom (61) and (62), we seek and such thatthat is,This impliesChoosing givesThen,Since then for we haveApplying (63), (67), (68), and (69) into (61), we getIt shows that This impliesorEquating (56) and (71) and solving for , we getAdding both sides of (56) and (71) givesUsing (73) into (74), we obtainSince this impliesApplying (56) and (71) into (76) and solving for , we getSince we must haveThus This completes the proof of Theorem 2.

Remark. Using (53), (54), and (56), it can be shown that and similarly by using (68), (69), and (71), it can be shown that
Now we are in the position to prove the characteristic C4. The following theorem shows that there exists such that

Theorem 3. Let be defined in (33), where , and be given. Then, there exists a constant such thatwhich is equivalent to

Proof. Since (33) becomesSince (33) becomesLetting we getComparing the denominators (80) and (82), we seek such thatObserve that (84) is just the square of (83). Note that Theorem 2 gives
Hence, (83) becomesThis yieldsThus, (82) simplifies toHence,Since This completes the proof of Theorem 3.

#### 5. Conclusion

In this paper, we have presented a boundary integral equation related to the Ahlfors map for a doubly connected region and an analytical method for finding the second zero of the Ahlfors map for an annulus region. The method does not depend on the series representation of the Szegö kernel. The method is based on some geometrical properties and knowledge of intersection points. Probably the method can be extended to some other doubly connected regions.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This study was supported partially by the Ministry of Education Malaysia (MOE) through the Research Management Centre (RMC), Universiti Teknologi Malaysia, GUP Ref. No. Q.J130000.2426.16H70. The first author would like to acknowledge the Ministry of Education Malaysia (MOE) for MyBrainSc scholarship.