The squeezing function on doubly-connected domains via the Loewner differential equation

For any bounded domains $\Omega$ in $\mathbb{C}^{n}$, Deng, Guan and Zhang introduced the squeezing function $S_\Omega (z)$ which is a biholomorphic invariant of bounded domains. We show that for $n=1$, the squeezing function on an annulus $A_r = \lbrace z \in \mathbb{C} : r<|z|<1 \rbrace$ is given by $S_{A_r}(z)= \max \left\lbrace |z| ,\frac{r}{|z|} \right\rbrace$ for all $0<r<1$. This disproves the conjectured formula for the squeezing function proposed by Deng, Guan and Zhang and establishes (up to biholomorphisms) the squeezing function for all doubly-connected domains in $\mathbb{C}$ other than the punctured plane. It provides the first non-trivial formula for the squeezing function for a wide class of plane domains and answers a question of Wold. Our main tools used to prove this result are the Schottky-Klein prime function (following the work of Crowdy) and a version of the Loewner differential equation on annuli due to Komatu. We also show that these results can be used to obtain lower bounds on the squeezing function for certain product domains in $\mathbb{C}^{n}$.


Introduction
In 2012, Deng, Guan and Zhang [8] introduced the squeezing function of a bounded domain Ω in C n as follows. For any z ∈ Ω, let F Ω (z) be the collection of all embeddings f from Ω to C n such that f (z) = 0. Let B(0; r) = {z ∈ C n : z < r} denote the n-dimensional open ball centered at the origin 0 with radius r > 0. Then the squeezing function S Ω (z) of Ω at z is defined to be Remark: (1) For the supremum in the definition of the squeezing function, we can restrict the family F Ω (z) to the subfamily of functions f such that f (Ω) is bounded. (2) For any λ = 0, we have f ∈ F Ω (z) if and only if λf ∈ F Ω (z). As a consequence, we may assume that b = 1.
It is clear from the definition that the squeezing function on Ω is positive and bounded above by 1. Also, it is invariant under biholomorphisms, that is, S g(Ω) (g(z)) = S Ω (z) for any biholomorphism g of Ω. If the squeezing function of a domain Ω is bounded below by a positive constant, i.e., if there exists a positive constant c such that S Ω (z) ≥ c > 0 for all z ∈ Ω, then the domain Ω is said to be holomorphic homogeneous regular by Liu, Sun and Yau [23] or with uniform squeezing property by Yeung [27]. The consideration of such domains appears naturally when one applies the Bers embedding theorem to the Teichmüller space of genus g hyperbolic Riemann surfaces.
The squeezing function is interesting because it provides some geometric information about the domain Ω. For instance, Joo and Kim proved in [19] that if Ω ⊂ C 2 is a bounded domain with smooth pseudoconvex boundary and if p ∈ ∂Ω is of finite type such that lim Ω z→p S Ω (z) = 1, then ∂Ω is strictly pseudoconvex at p. For another instance, Zimmer showed in [28] and [29] that if Ω ⊂ C n is a bounded convex domain with C 2,α boundary and K is a compact subset of Ω such that S Ω (z) ≥ 1 − for every z ∈ Ω \ K and for some positive constant = (n), then Ω is strictly pseudoconvex. In addition to providing geometric information, the squeezing function is related to some estimates of intrinsic metrics on Ω. For example, in [9], Deng, Guan and Zhang showed that for any point z in Ω and for any tangent vector v ∈ T z Ω, where C Ω and K Ω denote the Carathéodory seminorm and Kobayashi seminorm on Ω respectively. For other properties and applications of squeezing functions, see [8, 13-16, 20, 24, 25, 30].
Given a bounded domain Ω ⊂ C n , it is then natural to ask whether one can estimate or even compute the precise form for the squeezing function S Ω (z) on Ω. In [1], Arosio, Fornaess, Shcherbina and Wold provided an estimate of S Ω (z) for Ω = P 1 \K where K is a Cantor set. In [8], Deng, Guan and Zhang showed that the squeezing functions of classical symmetric domains are certain constants (using a result of Kubota in [22]); they also showed that the squeezing function of the n-dimensional punctured unit ball B(0; 1) \ {0} is given by S B(0;1)\{0} (z) = z .
We now consider the n = 1 case, and introduce the following notation which will be used in this paper. D r = {z ∈ C : |z| < r}, the disk of radius r centered at 0, and D = D 1 ; C r = {z ∈ C : |z| = r}, the circle of radius r centered at 0; and A r = {z ∈ C : r < |z| < 1}, the annulus with inner radius r and outer radius 1.
In this paper, we will disprove this conjecture by establishing the formula for S Ar (z). This also answers a question asked by Wold about the precise form for S Ar (z) in his lecture given in the Mini-workshop on Complex Analysis and Geometry at the Institute for Mathematical Sciences, NUS in May 2017.
Theorem 1.1. For 0 < r < 1, and r < |z| < 1, Remark: (1) The case of the punctured disk A 0 = D \ {0} follows by letting r → 0 so that S A 0 (z) = |z|. This is the n = 1 case of the result of Deng, Guan and Zhang for the punctured ball in C n referred to above. (2) Since any doubly-connected domain (other than the punctured plane) is conformally equivalent to A r for some 0 ≤ r < 1, this result determines the squeezing function in the doubly-connected case up to biholomorphisms.
Let us define By restricting F Ar (z) to F Ar (z), we will see in Section 4 that Theorem 1.1 will follow if we show that To do this, we will identify a candidate for the extremal function in F r (z). This will be the conformal map from A r onto a circularly slit disk, that is a domain of the form D \ L where L is a proper subarc of the circle with radius R ∈ (0, 1) and center 0. Through the results of Crowdy in [5] and [6], this conformal map can be expressed explicitly in terms of the Schottky-Klein prime function (see Theorem 2.1). It will be shown that, in this case, the radius of the slit is |z|. Then the following theorem will show that this conformal map is indeed extremal (which has been suggested by Wold in his lecture just mentioned).
Theorem 1.2. Let E ⊂ D be a closed set with 0 / ∈ E and there exists some constant y > 0 such that |z| ≥ y for any z ∈ E. Furthermore assume that Ω = D \ E is doubly connected. If g is a conformal map of A r onto Ω, for some r ∈ (0, 1), such that g maps the ∂D onto ∂D, then we have To prove this result, we will start with the case where E is a circular arc so that Ω = D \ E is a circular slit disk. We will then grow a curve from the circular slit so that E is now the union of the curve with the circular slit; the conformal maps onto these domains Ω will then satisfy a version of the Loewner differential equation due to Komatu (see [17,21]). Studying this differential equation will enable us to prove Theorem 1.2. The remaining cases for E then follow by letting the length of the circular slit tend to 0.
Finally, we obtain the following lower bound for the squeezing function on product domains in C n . Theorem 1.3. Suppose that Ω ⊂ C n and Ω = Ω 1 × · · · Ω n where Ω i is a bounded domain in C for each i. Then for z = (z 1 , · · · , z n ) ∈ Ω, we have Remark: The argument we use to prove the above theorem can be modified to obtain a similar result when Ω i are not necessarily planar. In [8], Deng, Guan and Zhang show that this inequality is attained in the case when each Ω i is a classical symmetric domain. Theorem 1.3 allows us to use the formula given in Theorem 1.1 for the squeezing function of a doubly-connected domain to get a lower bound on the squeezing function of the product of several doubly-connected and simply-connected domains. For example, considering Ω = A r × D, Theorem 1.3 together with Theorem 1.1 yields Obtaining the exact form for S Ar×D (z) would be of interest. The rest of the paper is organised as follows. Firstly, in Section 2, we review some results and concepts that are necessary for this paper including the formula for the conformal map of an annulus A r to a circularly slit disk D \ L in terms of the Schottky-Klein prime function and a version of the Loewner differential equation that we will need. Then, in Section 3, we give a proof for Theorem 1.2; the proof of Theorem 1.1 is provided in Section 4 and we prove Theorem 1.3 in Section 5. Finally, we discuss the multiply-connected cases in Section 6.

2.1.
Basic Definitions and Notations. Throughout this paper, we will make use of the following definitions and notations: • For z ∈ C n and r > 0, B(z; r) denotes the open ball centered at z with radius r. When n = 1, we also set D r = B(0; r) and in particular, D = B(0; 1). Then C r = ∂D r . • Let p > 0 and r = e −p so that 0 < r < 1. Then A r denotes the annulus centered at 0 with inner radius r and outer radius 1. In this case, A r is said be of modulus p. • Let Ω be a doubly-connected domain in C. The modulus p of Ω is defined to be the unique positive real number p such that there exists a biholomorphism φ from Ω to A r where r = e −p . • By a (doubly-connected) circularly slit disk, we refer to a domain of the form D \ L where L is a proper closed subarc of the circle with radius R ∈ (0, 1).
• For any set E ⊂ C, ∂E denotes the topological boundary of E in C.
• Let γ : I → C be a curve where I is an interval in R. We will write γI instead of γ(I) for notational simplicity. • We assume that the argument function Arg takes values in [0, 2π).
Recall from the previous section that a circularly slit disk is a domain of the form D \ L where L is a proper subarc of the circle with radius R ∈ (0, 1) and center 0. In [6], Crowdy established the following result.
Then f (·, y) is a conformal map from A r onto a circularly slit disk with f (∂D) = ∂D and y is mapped to 0.
See also [7]. Theorem 2.1 allows us to compute the radius of the circular arc in f (A r ). For any z = re iθ , we have .
This shows that the radius of the circular arc in f (A r ) is |y| and, in particular, it does not depend on r. Note that if φ is a conformal map from a circularly slit disk Ω 1 to another circularly slit disk Ω 2 such that φ maps ∂D to ∂D and φ(0) = 0, then φ must be a rotation (Lemma 6.3 in [4]). We restate this result as the following lemma.
Lemma 2.2. For any annulus A r and for any circularly slit disk Ω whose arc has radius y, if f is a conformal map which maps A r onto Ω with f (∂D) = ∂D, then we have Remark: We thank the referee for informing us that Lemma 2.2 is the same as Lemma 3 of [26]. Our proof of the above lemma is different from that of Lemma 3 in [26].
When y is positive, we have the following lemma.

Lemma 2.3.
Let Ω = D \ L be a circularly slit disk such that L is symmetric across the real axis, that is, z ∈ L if and only if z ∈ L. Let y be a point on an annulus A r such that r < y < 1. Let g : A r → Ω be the conformal map from A r onto Ω such that g(y) = 0 and g(∂D) = ∂D. Then for any z ∈ A r , g(z) = g(z).
In particular, if p 1 and p 2 denote the two end points of L and π 1 , π 2 ∈ C r denote the two points on the inner boundary of A r such that g(π 1 ) = p 1 and g(π 2 ) = p 2 ; then we have π 1 = π 2 .
Proof. Consider the map G defined by G(z) = g(z). Then G is a conformal map of A r onto Ω with G(∂D) = ∂D and G(y) = 0. Hence, G −1 • g is a conformal automorphism of A r that fixes y and does not interchange the boundary components. Since any such conformal automorphism of an annulus is the identity mapping, this implies that G −1 • g is the identity and so g(z) = g(z) on A r . Let {p i,n } ⊂ Ω be sequences of points such that p 2,n = p 1,n and lim It follows that for each n ∈ N, g(π 2,n ) = p 2,n = p 1,n = g(π 1,n ) = g(π 1,n ).
2.3. Approximating by Slit Domains. Let D be the collection of doubly-connected domains Ω so that Ω ⊂ A r for some r > 0 and ∂D be one of the boundary components of Ω. Let {Ω n } be a sequence in D. Define the kernel Ω of the sequence {Ω n } as follows: • If there exists some Ω * ∈ D such that Ω * ⊂ ∞ n=1 Ω n , then the kernel Ω is defined to be the maximal doubly-connected domain in D such that for any compact subset K of Ω, there exists an N ∈ N so that K ⊂ Ω n whenever n > N ; • otherwise, the kernel Ω is defined to be ∂D.
Then a sequence {Ω n } converges to Ω in the sense of kernel convergence, if Ω is the kernel of every subsequence of {Ω n }. Let {Ω n } be a sequence of doublyconnected domains in D. Since every doubly-connected domain of finite modulus is conformally equivalent to an annulus A r for some r ∈ (0, 1), there exists a sequence {ψ n } of conformal maps such that ψ n maps A rn onto Ω n and ψ n is normalized appropriately. A version of the Carathéodory kernel convergence theorem for doublyconnected domains will show that the kernel convergence of {Ω n } implies local uniform convergence of {ψ n }. This is Theorem 7.1 in [17]. We restate this result in a form which we will need later. Theorem 2.4. Suppose that r > 0 and r < y < 1. Let {Ω n } be a sequence of doubly connected domains in D such that y ∈ ∞ n=1 Ω n . Let {r n } be a sequence with r < r n < 1 for n ≥ 1 such that there exists a conformal map Φ n of Ω n onto A rn satisfying Φ n (y) > 0 and Φ n (∂D) = ∂D for every n. Then the kernel convergence of Ω n to a doubly connected domain Ω in D implies that the sequence {r n } converges to r and that the sequence {Φ n } converges locally uniformly to a conformal map Φ of Ω onto A r satisfying Φ(y) > 0 and Φ(∂D) = ∂D.
Theorem 2.4 can be obtained from Theorem 7.1 in [17] by renormalising the conformal maps. This theorem leads to the following proposition.
Assume there exists some y ∈ A r \ E with y > 0 and let Φ be the conformal map from A r \ E onto some annulus A r normalized such that Φ(y) > 0 and Φ(∂D) = ∂D.
(1) Suppose that ∂E ∩ A r is a Jordan arc. Then we can find a Jordan arc For the cases where ∂E ∩ A r is not a Jordan arc, we can find an increasing sequence {q n } with r < q n < 1 for all n and q n → r as n → ∞; a sequence of Jordan arcs Γ n ⊂ A r \ {y} which starts from C r ; conformal maps Φ n which map A r \ Γ n onto the annulus A qn with Φ n (y) > 0 and Φ n (∂D) = ∂D such that Φ n → Φ locally uniformly as n → ∞.
Proof. For part 1, suppose that ∂E ∩ C r = ∅, then we can find a Jordan arc γ : Then E is bounded by C r and a closed curve in A r . In this case, we define a Jordan arc γ : The conformal equivalence of any doubly-connected domains to an annulus implies that there exists an increasing function q : [0, T ] → [r, r ] with q(0) = r and q(T ) = r and a family of conformal maps Φ t of Ω t onto A q(t) with Φ t (y) > 0 and Φ t (∂D) = ∂D. Then as t → T , Ω t → A r \ E in the sense of kernel convergence. Hence, by Theorem 2.4, the sequence {Φ t } converges locally uniformly to a conformal map Φ of A r \ E onto A r such that Φ(y) > 0 and Φ(∂D) = ∂D. This proves part 1.
For part 2, since A r \ E is doubly connected, there exists an annulus A s and a conformal map f of A s onto A r \ E for some s > 0 such that f (∂D) = ∂D. For any 0 < δ < 1 − s, we let is an analytic Jordan arc. So part 1 of the proposition applies to A r \ E δ . That is, we can find a Jordan arc γ δ : Letting δ → 0, and applying a diagonal argument we get the desired result. More precisely, let {t n } ⊂ [0, T ) be an increasing sequence and the desired result will follow by letting The simply-connected version of this proposition is given in Theorem 3.2 in [11]. This proposition allows us to consider slit domains in the proof of Theorem 1.2 via a Loewner-type differential equation which is introduced in the next section. This is analogous to the approach to solving various coefficient problems for univalent functions on D (including the De Branges-Bieberbach Theorem); see [11] and references therein.

The Loewner-type Differential Equation in
Annuli. In this section, we introduce the Loewner differential equation which is a differential equation for the conformal maps (normalized and parametrized appropriately) onto a slit domain. We first introduce the classical setting in annuli, where the slit grows from the outer boundary.
For p > 0 , define r = e −p and r t = e −p+t . Suppose that γ : Then there exists a family of conformal maps φ t : and φ t (z) satisfies the Komatu's version of the Loewner Differential Equation on an annulus (see [17]), Here, for α ∈ ∂D and r > 0, For our purposes, we need a version of Loewner-type differential equations where the curve grows from the inner boundary circle of A r 0 . Let C r 0 be the circle centered at 0 with radius r 0 . Suppose that γ : which is a conformal automorphism of A rt that interchanges inner boundary and outer boundary of A rt . Hence ρ t • γ is a Jordan arc satisfying the conditions given at the beginning of this subsection and for this Jordan arc ρ t • γ, let φ t be the corresponding conformal map satisfying (5). Then we define By the chain rule, Φ t satisfies the differential equation where β(t) satisfies Φ t (γ(t)) = r t e iβ(t) ∈ A rt . Let y 0 > 0 be a fixed point in A r 0 . By composing Φ t with a suitable rotation, we can normalize Φ t such that y t := Φ t (y 0 ) > 0 for any t. Then Φ t satisfies , e −iβ(t) + iJ(r t , y t , β(t)) for some function J(r t , y t , β(t)). Notice that the normalization accounts for multiplying a rotation factor to Φ t at each time t. So the function J(r t , y t , β(t)) is real-valued. Also, since y t > 0 for any t, we have ∂ t Im (log y t ) = 0 and hence we have J(r t , y t , β(t)) = Im K rt r t y t , e −iβ(t) .
We call the function β(t) the Loewner driving function of the curve γ. We now define Hence Substituting z = y 0 in the above equation and noting that y t = Φ t (y 0 ) > 0, we get ∂ t log y t = P (r t , y t , β(t)) where P (r, y, θ) := R(r, θ, y) = Re 1 − lim for 0 < r < y < 1.
We now make the following construction which is illustrated in Figure 1.

3.1.
Idea of the Proof. Suppose that y > 0. Let E ⊂ D be a closed set with 0 ∈ E and |z| ≥ y for all z ∈ E. Let g be a conformal map of an annulus A r onto D \ E. By further composing with a rotation we can assume that g −1 (0) > 0. We need to show that g −1 (0) ≥ y.
We first consider the case where E is the union of a circular arc L (with radius y centred at 0) and a Jordan arc starting from L. Proposition 2.5 will allow us to obtain the general case by an approximation argument.
Denote by f the conformal map of the annulus A r 0 onto a circularly slit domain D \ L which maps a point y ∈ A r 0 with y > 0 to 0. Let γ be a Jordan arc growing from the circular arc L. In other words, γ : [0, T ] → D is a Jordan arc satisfying γ(0) ∈ L and γ(0, T ] ⊂ Ω, such that Ω \ γ(0, t] has modulus − log r t = −(log r 0 ) − t. Now, we let γ = f −1 • γ. Let y 0 = y and we then define y t and β(t) as in Section 2.4. Now let f t be the conformal map of the annulus A rt onto a circularly slit domain where L t is a circular arc, which maps the point y t ∈ A rt to 0. The maps f t can each be extended continuously to the inner circle C rt of A rt and f t maps C rt onto the circular arc L t . The preimages under f t of the two endpoints of the circular arc L t then partition C rt into two circular arcs which are symmetric under the transformation z → z according to Lemma 2.3. We call these circular arcs Γ + t and Γ − t respectively, where Γ + t is the circular arc which intersects the negative real axis.
It can be shown that β(t) ∈ Γ + t for all t ∈ [0, T ] implies that y t is strictly increasing. Similarly, β(t) ∈ Γ − t for all t ∈ [0, T ] implies that y t is strictly decreasing. It would thus be sufficient to show that if | γ(t)| > y, then β(t) ∈ Γ + t for all t ∈ [0, T ]. However, in general, this may not be the case. The idea of our method is as follows: Since | γ(t)| > y, we can extend the length of L to a longer circular arc L * 0 without ever intersecting γ. As the curve γ grows from L * 0 , we simultaneously shrink L * 0 to L * t such that at time T , L * T coincides with L. By choosing a suitable rate at which L * t shrinks to L from each end of the circular arc L * 0 , we will be able to show that the preimage of 0 at each t, y * t , is now strictly increasing. This will prove the desired result since y * T = y T (as L * T = L ) and y * 0 = y 0 . The second equality follows from the fact that extending the length of L to get L * 0 does not change the preimage of 0 by Lemma 2.2. It is for this part of the argument that we will need to use the three-slit Loewner differential equation from Section 2.5: the three slits will be the curve γ, the circular arc in the clockwise direction and the circular arc in the anticlockwise direction.
The rest of this section provides the formal construction of the above argument.

Properties of P (r, y, θ) in the Loewner-Type Differential Equation.
In this subsection, we study properties of the differential equation in (7). The following lemma gives some properties of P (r, y, θ) that we will need.
Proof. As y > 0, we can see that Taking real parts of the above equation proves part 1.
To prove part 2, we write r = e −p , p > 0 and define This function A is related to the Jacobi theta function ϑ 4 defined in Section 13.19 of [12]. Direct calculations show that P (r, y, θ) = 2Im A (θ + i ln y; p) A(θ + i ln y; p) where A (z; p) = ∂ z A(z; p). It then follows that ∂ θ P (r, y, θ) = 2Im A (θ + i ln y; p)A(θ + i ln y; p) − (A (θ + i ln y; p)) 2 (A(θ + i ln y; p)) 2 . Define Then G 1 (z; p) is an elliptic function (of z) with periods 2π and 2ip. Note that G 1 has poles of order 2 at z = 2nπ + (2m − 1)ip for any n, m ∈ Z and these are the only poles of G 1 . Let ℘ be the Weierstrass's ℘ function with period 2π and 2ip. It follows that there exists an constant c 1 such that G 2 (z) := G 1 (z; p) − c 1 ℘(z + ip) has no pole on C. Thus G 2 (z) = c 2 for some constant c 2 by the Liouville's theorem. This means that we have for some constant c 1 , c 2 . By considering the Laurent series expansions of G 1 and ℘ and comparing coeffients, we have c 1 = −1 and c 2 is real, that is, for some real constant c. (For details, see the demonstration by Dixit and Solynin [10] for equation (3.3) of [10].) Hence, ∂ θ P (r, y, θ) = −2Im (℘ (z + ip)) .
As a consequence of Lemma 3.1, we have the following lemma.
The above proposition allows us to prove Theorem 1.2.
Proof of Theorem 1.2. Define E min = {z ∈ E : |z| ≤ |w| for any w ∈ E}. So E min consists of all the points in E closest to the origin. Thus, E min is the union of circular arc(s) with the same radius y 0 ≥ y and clearly E min ⊂ ∂E. Note that either E min = E or E min E.
If E min = E, then the connected set E is a circular arc centered at 0. In this situation, Lemma 2.2 implies that |g −1 (0)| = y 0 .
If E min E, then there are two subcases: either there exists a connected component L of E min containing more than one point or E min is a set of disconnected points.
Suppose that there is a connected component L of E min containing more than one point. We first assume that E \ L is a Jordan arc γ. Then we can find 0 < r 0 < 1 such that A r 0 has the same modulus as the domain D \ L. Let y 0 > r 0 and let f (·, y 0 ) be the conformal map of A r 0 onto D \ L with f (y 0 , y 0 ) = 0. Define γ : [0, T ] → A r 0 to be the Jordan arc such that γ(0) ∈ C r and the image of γ(0, T ] under f (·, y 0 ) is γ. In addition, γ is parametrized such that A r 0 \ γ(0, t] has modulus −(log r 0 ) + t. We define y t as in Section 2.4, namely y t = Φ t (y 0 ) where Φ t is a conformal map from A r 0 \ γ(0, t] onto A rt . Note that Φ T = Φ τ (T ) and hence y T = y τ (T ) . Then by Proposition 3.4, y T > y 0 . Since g −1 (0) = y T , we have |g −1 (0)| > y 0 . The case where E \ L is not a Jordan arc follows from part 2 of Proposition 2.5.
The final case where E min is a set of disconnected points follows from the previous case by letting the arc length of L shrink to 0.

Proof of the Main Result
Recall that the squeezing function is defined to be where F Ar (z) = {f : f is a conformal map from A r to C such that f (z) = 0.} .
To simplify notation, we write S r (z) = S Ar (z) and F r (z) = F Ar (z). We have the following corollary of Theorem 1.2. Proof.
The conformal map f of A r onto a circularly slit disk with z mapping to 0 is in F r (z). Lemma 2.2 implies that S r (z) ≥ |z|. Now suppose that we can find f * ∈ F r (z) such that D a * ⊂ f * (A r ) for some a * > |z|. Let E be the bounded component of the complement of f * (A r ) in C. Then |w| ≥ a * for all w ∈ E. Theorem 1.2 implies that, |(f * ) −1 (0)| ≥ a * which is a contradiction since (f * ) −1 (0) = z.
It now remains to prove Theorem 1.1. For any bounded doubly-connected domain Ω, ∂Ω has two connected components: we denote the component that separates Ω from ∞ (i.e. the outer boundary) by ∂ o Ω; we denote the other component (i.e. the inner boundary) by ∂ i Ω. We decompose the family F r (z) into two disjoint subfamilies F 1 r (z) consists of functions that map outer boundary to outer boundary; F 2 r (z) consists of functions that interchange inner and outer boundary. We will consider a squeezing function on each subfamily separately. Define Then the squeezing function satisfies S r (z) = max{S 1 r (z), S 2 r (z)}. Proof. This follows from the fact that f ∈ F 1 r (z) if any only if f • ρ ∈ F 2 r ( r z ) where ρ(z) = r z .
Proof of Theorem 1.1. By Corollary 4.1 and Lemma 4.2, it is sufficient to prove that S 1 r (z) = S r (z). First we note that F r (z) ⊂ F 1 r (z) and hence S r (z) ≤ S 1 r (z). Since F 1 r (z) is a normal family of holomorphic functions, it follows easily that we can replace the sup in the definition of S 1 r (z) with max. Let f ∈ F 1 r (z) be any function that attains this maximum with corresponding a i.e.
D a ⊂ f (A r ) ⊂ D and a = S 1 r (z).
We denote by Ω the simply-connected domain satisfying 0 ∈ Ω and ∂Ω = ∂ o f (A r ). Note that D a ⊂ Ω ⊂ D. By the Riemann mapping theorem, we can find a conformal map g of D onto Ω such that g(0) = 0. Let F = g −1 • f . Then F ∈ F r (z) and In addition, g maps the unit disk into itself and so by the Schwarz lemma, g(D a ) ⊂ D a . Combining this with the above, we deduce that Hence a ≤ sup{ρ : D ρ ⊂ F (A r )} which implies that S 1 r (z) ≤ S r (z). Therefore S 1 r (z) = S r (z) as required.
5. The squeezing function on product domains in C n It remains to prove Theorem 1.3.
Proof of Theorem 1.3. The squeezing function is scale invariant and hence in the definition of the squeezing function S Ω i (z i ), we can restrict the family F Ω i (z i ) to F b Ω i (z i ) = {f ∈ F Ω i (z i ) : |f (w)| < 1 for all w ∈ Ω i }.
Since F b Ω i (z i ) is a normal family, one can easily show that there is a function f i in F b Ω i (z i ) attaining the supremum in the definition of S Ω i (z i ). By scaling, we can assume also that sup{|f i (w)| : w ∈ Ω i } = 1. Consider λ i = S −1 Ω i (z i ) and g(w) = (g 1 (w 1 ), · · · , g n (w n )) where g i = λ i f i . Since g i are holomorphic and injective for all i, g(w) is a holomorphic embedding of Ω into C n . Also, f i (z i ) = 0 for each i and thus g ∈ F Ω (z). Moreover, since f i attains the supremum in S Ω i (z i ), we have D λ −1 i ⊂ f i (Ω i ) ⊂ D and hence D ⊂ g i (Ω i ) ⊂ D λ i . It follows that D n ⊂ g(Ω) ⊂ D λ 1 × · · · × D λn .

The squeezing function on multiply-connected domains
We now discuss some future directions regarding the squeezing function on planar domains of higher connectivity.
Let Ω ⊂ C be a finitely connected domain with disjoint boundary components γ 0 , γ 1 , · · · γ n . As a corollary of conformal equivalence for finitely connected regions [4], every finitely connected domain with non-degenerate boundary components is conformally equivalent to a circular domain, (i.e., the unit disk with smaller disks removed). Thus we can assume that γ 0 is the unit circle and γ 1 , . . . , γ n are circles contained inside the unit disk. In light of our results, for a fixed z ∈ Ω, we propose that the function which attains the maximum in the extremal problem : is given by the conformal map of Ω onto a circularly slit disk of the same connectivity (i.e. the unit disk with proper arcs of circles centered at 0 removed). For j = 1, . . . , n, let µ j be a Möbius transformation that interchanges γ j and the unit circle ∂D and let Ω j = µ j (Ω). We also let µ 0 be the identity mapping and Ω 0 = Ω. Then, for j = 0, . . . , n, let f j denote the conformal map of Ω j onto a circularly slit disk with f j (µ j (z)) = 0 and let Rad(Ω j ) be the minimum of the radii of the circular arcs in f j (Ω j ). Note that Rad(Ω j ) does not depend on the choice of µ j or f j (by Theorem 6.2 in [4]). We make the following conjecture regarding the squeezing function on Ω.
The Schottky-Klein prime function ω(·, ·) can be defined on domains of connectivity n in terms of the Schottky Group of Ω (see [6]). By [5], the same expression in Theorem 2.1, f (·, z) = ω(·, z) |z|ω(·, z −1 ) gives the formula for the conformal map of Ω onto a circularly slit disk mapping z to 0. In addition, an expression for Rad(Ω) is also given in [5].
Recently, Böhm and Lauf [3] have obtained an expression for a version of the Loewner differential equation on n-connected circularly slit disks. Using Crowdy's version of the Schwarz-Christoffel formula for multiply-connected domains (given in [5]), it should be possible to express the Loewner differential equation in terms of the Schottky-Klein prime function. It is anticipated that the methods in our paper could then be used to prove this conjecture.