aa r X i v : . [ m a t h . C V ] M a r A GENERALIZED HURWITZ METRIC
ARSTU † AND SWADESH KUMAR SAHOO † ∗
Abstract.
In 2016, the Hurwitz metric was introduced by D. Minda in arbitrary propersubdomains of the complex plane and he proved that this metric coincides with the Poincar´e’shyperbolic metric when the domains are simply connected. In this paper, we provide analternate definition of the Hurwitz metric through which we could define a generalized Hur-witz metric in arbitrary subdomains of the complex plane. This paper mainly highlightsvarious important properties of the Hurwitz metric and the generalized metric including thesituations where they coincide with each other. Introduction and Preliminaries
In 19 th century, the notion of hyperbolic metric was first introduced. As pointed out,for instance in [5, p. 132] and [8], the hyperbolic density on a hyperbolic domain Ω canbe understood through the extremal problem of maximizing | f ′ (0) | over all holomorphicfunctions f that map the unit disk into Ω. In 1981, Hahn [2] introduced a pseudo-differentialmetric for complex manifolds by means of an extremal problem. Two years later, Minda [7]reconsidered the Hahn metric in Riemann surfaces. Recently, Minda considered an extremalproblem of Hurwitz [3] and introduced a new conformal metric, namely, the Hurwitz metric[8] in any proper subdomain of the complex plane C . Our objective in this paper is toinvestigate further properties of the Hurwitz metric and their applications.In 2007, Keen and Lakic [4] defined some new densities in arbitrary plane domains thatgeneralize the hyperbolic density. They are namely the generalized Kobayashi density (see [4,Definition 2]) and the generalized Carath´eodory density (see [5, Definition 9.2, p. 166]). TheKobayashi density is defined by pushing forward the hyperbolic density from the unit disk toa plane domain by a holomorphic function, whereas, the Carath´eodory density is defined bypulling back the hyperbolic density from a plane domain to the unit disk by a holomorphicfunction. This paper deals with a generalized Hurwitz metric in the sense of Kobayashi. Weare considering a generalized Hurwitz metric in the sense of Carath´eodory in our next paper.Since holomorphic functions are infinitesimal contractions in the hyperbolic metric, it is easyto see that the generalized Kobayashi density exceeds over hyperbolic density on hyperbolicdomains. Furthermore, the hyperbolic and the generalized Kobayashi densities coincide Mathematics Subject Classification.
Primary: 30F45; Secondary: 30C20, 30C80.
Key words and phrases.
Hyperbolic metric, Kobayashi metric, Hurwitz metric, Hurwitz covering, gener-alized Hurwitz metric, hyperbolic domain, Lipschitz domain.* The corresponding author. whenever there is a regular holomorphic covering map [5, p 125] from the source domainto the range domain. Similar to the case of hyperbolic distance the generalized Kobayashidistance, in association with the generalized Kobayashi density, between two points can bedefined by taking infimum of the generalized Kobayashi length of all rectifiable paths joiningthe points. In fact, with this definition, it becomes a complete metric space.As an analogue of the generalized Kobayashi density, we shall generalize the Hurwitzmetric by pushing forward the Hurwitz density from a proper subdomain of the complexplane to an arbitrary domain by holomorphic functions with some specific properties. Wecall this new density the generalized Hurwitz density . Note that, on hyperbolic domains theHurwitz density exceeds the hyperbolic density. Furthermore, in this work we prove thatthe generalized Hurwitz density is always greater than the Hurwitz density.Throughout this article, our notations are relatively standard and we are working mainlyon the complex plane C . First we denote the open unit disk by D := { w ∈ C : | w | < } . Theclassical Hyperbolic density [5, p. 33] in D is defined as λ D ( w ) = 21 − | w | for w ∈ D . Note that we consider the hyperbolic metric with constant curvature −
1. Thehyperbolic distance between two points w , w in D is λ D ( w , w ) = inf Z γ λ D ( w ) | dw | ,where the infimum is taken over all paths γ joining w and w in D . Since the hyperbolicmetric is conformal invariant, by the Riemann Mapping Theorem one can easily define iton the proper simply connected domains of C . However, this metric is also defined in moregeneral domains so-called hyperbolic domains. A plane domain Ω is called hyperbolic if C \ Ω contains at least two points. On a hyperbolic domain Ω, the hyperbolic density λ Ω [5,p. 124] is λ Ω ( w ) = λ D ( t ) | π ′ ( t ) | ,where π : D → Ω is a universal covering map with π ( t ) = w . Analogue to the case of theunit disk, the hyperbolic distance between two points w and w in Ω is defined by λ Ω ( w , w ) = inf Z γ λ Ω ( w ) | dw | ,where the infimum is taken over all paths γ joining w and w in Ω.We now define some notations that are used in the definition of the Hurwitz density definedin [8].Throughout this paper we denote by H ( Y , Ω) for the set of all holomorphic functions froma domain Y to another domain Ω. For a fixed point s ∈ D , let H s ( w , Ω) be the family ofall holomorphic functions h from D into Ω with h ( s ) = w , h ( t ) = w for all t ∈ D \ { s } and GENERALIZED HURWITZ METRIC 3 h ′ ( s ) >
0. For a point w ∈ Ω, we write H ( w , Ω) := H ( w , Ω). The Hurwitz density [8] in Ωis defined as η Ω ( w ) = 2 G ′ (0) = 2 r Ω ( w ) ,where G ′ (0) = max { h ′ (0) : h ∈ H ( w , Ω) } =: r Ω ( w ). If Ω ( C is a domain and b ∈ Ω isany point, then the covering map G : D \ { } → Ω \ { b } extends to a holomorphic function G b : D → Ω with G b (0) = b , G ′ b (0) >
0. The extended holomorphic function G b is called the Hurwitz covering [8] of ( D , 0) onto (Ω, b ).The distance decreasing property, which is stated below, of the Hurwitz density for theholomorphic function plays a crucial role to prove our results in this article. Theorem A (Distance decreasing property of the Hurwitz density). [8]
Suppose that Ω and △ are proper subdomains of C , a ∈ Ω and b ∈ △ . If h is a holomorphic function of Ω into △ with h ( a ) = b and h ( w ) = b for w ∈ Ω \ { a } , then η △ ( b ) | h ′ ( a ) | ≤ η Ω ( a ). Moreover, equality holds if and only if h is a covering of Ω \ { a } onto △ \ { b } that extendsto a holomorphic map of Ω onto △ with h ( a ) = b and h ′ ( a ) = 0.The Structure of this document is organized as follows. In Section 2, certain basic prop-erties of the Hurwitz density are studied leading to the concept of Hurwitz distance whichproduces the completeness property of the metric space. Furthermore, Section 3 deals withsome basic properties of the generalized Hurwitz density. Finally, in Section 4, we relate theHurwitz and the the generalized Hurwitz densities over some specific plane domains havingcertain geometric properties. 2. The Hurwitz Metric
Firstly, we present here a characterization of the Hurwitz density which gives us ideasto introduce the notion of generalized Hurwitz density in the next section. Let F be theextremal function for the extremal problem max { h ′ ( s ) : h ∈ H s ( w , Ω) } . Then, we have F ′ ( s ) = max { h ′ ( s ) : h ∈ H s ( w , Ω) } = max { h ′ ( s ) = ( f ◦ T ) ′ ( s ) : f ∈ H ( w , Ω) and T , the M¨obius transformationof D onto itself with T ( s ) = 0 and T ′ ( s ) > } .Since T ( z ) = ( z − s ) / (1 − sz ), it follows that F ′ ( s ) = max ( f ′ (0)1 − | s | : f ∈ H ( w , Ω) ) . ARSTU AND S. K. SAHOO
Since the hyperbolic density on D is given by λ D ( s ) = 2 / (1 − | s | ), by the notations definedin the previous section, we have F ′ ( s ) = λ D ( s ) η Ω ( w ).By using this argument, we provide here an alternate definition of the Hurwitz density asfollows: Definition 2.1.
The
Hurwitz density on a proper subdomain Ω of C is defined as(2.1) η Ω ( w ) = η D ( s ) g ′ ( s ) ,where g = h ◦ T such that T is the M¨obius transformation from D onto D with T ( s ) =0, T ′ ( s ) > h is the Hurwitz covering map from D onto Ω with h (0) = w .To define the Hurwitz distance between any two points in Ω we integrate the density η Ω and obtain the following definition: Definition 2.2. [Hurwitz distance] For w , w in Ω, we define η Ω ( w , w ) = inf Z γ η Ω ( w ) | dw | ,where infimum is taken over all rectifiable paths γ in Ω joining w and w .Note that we are using the same notation for the Hurwitz density as well as the Hurwitzdistance between any two points where the distinction can be observed by seeing the numberof parameters. However, now onward, for simplicity, we sometimes use the notation η Ω for η Ω ( w , w ). To justify our above definition we indeed prove that η Ω defines a metric whenthe domain Ω is assumed to be hyperbolic. Theorem 2.3. If Ω is a hyperbolic domain, then (Ω, η Ω ) is a complete metric space.Proof. By the definition of η Ω , symmetry and triangle inequality follow directly. Thereforeto prove that (Ω, η Ω ) is a metric space, we need to prove strictly positivity of the Hurwitzdistance between any two distinct points. Let w , w be any two distinct points in Ω. Since η Ω ( w , w ) is the infimum of the Hurwitz length of all rectifiable curves joining w and w inΩ, for any ǫ > γ such that η Ω ( w , w ) ≥ Z γ η Ω ( w ) | dw | − ǫ .Note that in a hyperbolic domain Ω, the inequality η Ω ≥ λ Ω is well-known; see [8]. Then wehave η Ω ( w , w ) ≥ Z γ λ Ω ( w ) | dw | − ǫ ≥ λ Ω ( w , w ) − ǫ . GENERALIZED HURWITZ METRIC 5
Letting ǫ →
0, we obtain(2.2) η Ω ( w , w ) ≥ λ Ω ( w , w ) > a locally compact length ( metric ) space X is complete if and only if every closed disc in X is compact ( see also [1, p. 28]). Because λ Ω is complete, each closed hyperbolic disk D λ Ω ( a , r ) = { w ∈ Ω : λ Ω ( a , w ) ≤ r } is compact. Because λ Ω ≤ η Ω , D η Ω ( a , r ) ⊂ D λ Ω ( a , r ).A closed subset of a compact set is compact, so D η Ω ( a , r ) is compact. (cid:3) The following remark assures that there exists a non-hyperbolic domain for which Theo-rem 2.3 still satisfies.
Remark 2.4.
Let Ω = C \ { } . Then, the Hurwitz density has the elementary formula η Ω ( w ) = 1 / | w | . This is nothing but a scalar multiplication of the classical quasihyperbolicmetric of Ω. The completeness property now follows from the fact that the quasihyperbolicmetric space is complete.We know that the holomorphic functions are global as well as infinitesimal contractionfunctions with respect to the hyperbolic metric. In analogy to this we now prove that theone-to-one holomorphic functions are global contraction functions for the Hurwitz metric aswell. Proposition 2.5.
Let Ω and Y be proper subdomains of C and h be an injective holomorphicfunction from Ω to Y . Then we have the inequality η Y ( h ( w , w )) ≤ η Ω ( w , w ) for all w , w in Ω. Equality holds in the above inequality if h is an conformal homeomor-phism.Proof. By definition of η Ω ( w , w ), for any ǫ > γ joining w and w inΩ such that Z γ η Ω ( w ) | dw | ≤ η Ω ( w , w ) + ǫ .By Definition 2.2, it follows clearly that(2.3) η Y ( h ( w , w )) ≤ Z h ( γ ) η Y ( z ) | dz | = Z γ η Y ( h ( w )) | h ′ ( w ) || dw | .Since h is one-to-one holomorphic function, by Theorem A, we have(2.4) η Y ( h ( w )) | h ′ ( w ) | ≤ η Ω ( w )for every w in Ω. Combining (2.3) and (2.4), we obtain η Y ( h ( w , w )) ≤ Z γ η Ω ( w ) | dw | ≤ η Ω ( w , w ) + ǫ . ARSTU AND S. K. SAHOO
Letting ǫ →
0, we conclude what we wanted to prove. (cid:3) The generalized Hurwitz Metric
In Section 2 we discussed the alternate definition of the Hurwitz density. By adoptingthe idea of generalized Kobayashi density we are going to define and study the generalizedHurwitz density in this section. The distance decreasing property of the Hurwitz densityimplies that for any holomorphic function h from D to Ω with h ( s ) = w , h ( t ) = w for all t in D \ { s } and h ′ ( s ) = 0, we have the inequalities η Ω ( h ( s )) | h ′ ( s ) | ≤ η D ( s ),and η Ω ( h ( s )) ≤ η D ( s ) | h ′ ( s ) | .Since the formula (2.1) provides an existence of a holomorphic function h for which theequality holds, we have η Ω ( w ) = inf η D ( s ) | h ′ ( s ) | ,where the infimum is taken over all holomorphic functions h from D to Ω with h ( s ) = w , h ( t ) = w for all t ∈ D \ { s } , h ′ ( s ) = 0. This leads to the notion of introducing generalizedHurwitz density for an arbitrary domain Ω. Definition 3.1.
For any domain Ω ⊂ C , the generalized Hurwitz density is defined as η D Ω ( w ) = inf η D ( s ) | h ′ ( s ) | ,where the infimum is taken over all h ∈ H ( D , Ω) with h ( s ) = w , h ( t ) = w for all t ∈ D \ { s } , h ′ ( s ) = 0, and all s in D . Remark 3.2.
If Ω ( C and a ∈ Ω, then there exists a Hurwitz covering map g from D toΩ which realizes the infimum; thus η Ω ( a ) = η D Ω ( a ) for all a ∈ Ω.Note that, in Definition 3.1 it is not required to choose the domain Ω to be a propersubdomain of the complex plane C . In the following theorem, we calculate η D Ω , when Ω = C . Theorem 3.3.
Suppose that Ω is the whole complex plane, then the generalized Hurwitzdensity η D Ω ( w ) is identically equal to zero for all elements w in Ω. Proof.
Let w ∈ Ω be an arbitrary element and n be any positive integer. Setting h n ( s ) =( s − t ) n + w . Clearly, h n is a sequence of holomorphic functions from D into C with h n ( t ) = w and h n ( s ) = w for all s ∈ D \ { t } . By Definition 3.1, we have η D Ω ( w ) ≤ η D ( t ) | h ′ n ( t ) | . GENERALIZED HURWITZ METRIC 7
Since the Hurwitz and the hyperbolic densities coincide on simply connected domains, wehave η D Ω ( w ) ≤ λ D ( t ) n .Letting n goes to infinity, we obtain that η D Ω ( w ) = 0. (cid:3) The definition of the generalized Hurwitz density can further be generalized by changingthe fixed domain D to an arbitrary proper subdomain Y of C , that is, by pushing forwardthe Hurwitz density on Y to Ω by a holomorphic function having some special property.Here, we call Y as the basepoint domain . This idea leads to the following definition. Definition 3.4.
Let Ω ⊂ C be arbitrary. For all s ∈ Y , the generalized Hurwitz density η Y Ω for the basepoint domain Y is defined as η Y Ω ( w ) = inf η Y ( s ) | h ′ ( s ) | ,where η Y is the Hurwitz density on Y and the infimum is taken over all holomorphic functions h from Y to Ω with h ( s ) = w , h ( t ) = w for all t ∈ Y \ { s } , h ′ ( s ) = 0.In view of the nature of Definition 3.4, it is here appropriate to remark that η Y Ω can be+ ∞ at some points, or even at every point.We will now prove some expected elementary properties of η Y Ω . We start by comparing theHurwitz and the generalized Hurwitz densities on proper subdomains of the complex plane. Proposition 3.5.
Let Y ⊂ C be a domain and Ω be a proper subdomain of C . Then forevery point w in Ω, we have η Y Ω ( w ) ≥ η Ω ( w ). Proof.
Let a ∈ Y and h be any holomorphic function from Y to Ω with h ( a ) = b , h ( s ) = b for all s ∈ Y \ { a } and h ′ ( a ) = 0. Then by distance decreasing property of the Hurwitzdensity, we have η Ω ( h ( a )) | h ′ ( a ) | ≤ η Y ( a ),and η Ω ( b ) ≤ η Y ( a ) | h ′ ( a ) | .Taking the infimum on both sides over h ∈ H ( Y , Ω) with h ( a ) = b , h ( s ) = b for all s ∈ Y \ { a } , h ′ ( a ) = 0, we have η Y Ω ( b ) ≥ η Ω ( b ).Since b ∈ Ω is arbitrary, the above inequality holds true for every b ∈ Ω. (cid:3) ARSTU AND S. K. SAHOO
One naturally asks the comparison between the classical generalized Kobayashi densityand the generalized Hurwitz density. Recall the definition of the generalized Kobayashidensity.
Definition 3.6.
Let Ω be a domain in the complex plane. For every w ∈ Ω, the generalizedKobayashi density is given by κ Y Ω ( w ) = inf λ Y ( t ) | f ′ ( t ) | ,where λ Y is the hyperbolic density on a hyperbolic domain Y ⊂ C and the infimum is takenover all f ∈ H ( Y , Ω) and all points t ∈ Y such that f ( t ) = w .We immediately have Corollary 3.7. If Y and Ω are hyperbolic domains, then η Y Ω ≥ κ Y Ω . Corollary 3.8.
For proper subdomains Ω and Y of C , we have η Y Ω ( w ) ≥ for all w in Ω.There are certain situations where the classical generalized Kobayashi density agrees withthe hyperbolic density, see for instance [4]. This motivates us to investigate the situationsunder which the generalized Hurwitz density coincides with the Hurwitz density. The fol-lowing proposition justifies one such case and a few more situations will be covered in thenext section.
Proposition 3.9.
Let Ω and Y be proper subdomains of C . If for every b ∈ Ω, there exists aholomorphic covering map h b from Y \ { a } onto Ω \ { b } that extends to a holomorphic mapof Y onto Ω with h b ( a ) = b and h ′ b ( a ) = 0, then η Y Ω ( w ) = η Ω ( w ) for all w in Ω. In particular, we also have η ΩΩ ( w ) = η Ω ( w ) for every w in Ω. Proof.
Since h b is a holomorphic map from Y to Ω with h b ( a ) = b , h b ( w ) = b for all w in Y \ { a } and h ′ b ( a ) = 0, by the definition of generalized Hurwitz density, we have(3.1) η Y Ω ( b ) ≤ η Y ( a ) | h ′ b ( a ) | .In addition, by Theorem A, we obtain(3.2) η Ω ( b ) | h ′ b ( a ) | = η Y ( a ).Combining (3.1) and (3.2), we obtain η Y Ω ( b ) ≤ η Ω ( b ). GENERALIZED HURWITZ METRIC 9
Since b is an arbitrary point, it follows that η Y Ω ( w ) ≤ η Ω ( w ) for all w ∈ Ω. On the otherhand, by Proposition 3.5 it follows that η Y Ω ( w ) ≥ η Ω ( w ). Hence the proof is complete. (cid:3) Proposition 3.9 is stronger, because for non-simply connected domains Y and Ω the propo-sition certainly holds (see for instance Example 3.12). To demonstrate this, we use thedistance decreasing property of the generalized Hurwitz density, which is proved below (seeTheorem 3.11). However, for simply connected domains we have the following special situ-ation. Corollary 3.10. If Y ( C is a simply connected domain and Ω ( C is any domain, then η D Ω ≡ η Y Ω ≡ η Ω . Proof.
Since Y ( C is a simply connected domain, by Riemann Mapping Theorem thereexists a conformal homeomorphism T from Y onto D . Furthermore, Ω ( C implies that forevery point w ∈ Ω there is a Hurwitz covering map g w from D onto Ω with g w (0) = w .Hence, by using the composed map g ◦ T from Y onto Ω in Proposition 3.9, we conclude ourresult. (cid:3) It is well-known that both the Hurwitz and the hyperbolic metrics as well the generalizedKobayashi metric κ have distance decreasing properties. The following result provides asimilar property for the generalized Hurwitz metric. Theorem 3.11. (Distance decreasing property of the generalized Hurwitz density)
Let Ω and △ be any subdomains of C and Y ( C be a domain. If h is a holomorphic functionfrom Ω to △ with h ( a ) = b , h ′ ( a ) = 0 and h ( w ) = b for all w ∈ Ω \ { a } , then η Y △ ( h ( a )) | h ′ ( a ) | ≤ η Y Ω ( a ). Proof.
By the definition of generalized Hurwitz density, for every ǫ > c ∈ Y and a holomorphic map g from Y to Ω with g ( c ) = a , g ( s ) = a for all s ∈ Y \{ c } , g ′ ( c ) = 0 and(3.3) η Y Ω ( a ) ≥ η Y ( c ) | g ′ ( c ) | − ǫ .Note that, h ◦ g maps Y to △ such that ( h ◦ g )( c ) = b , ( h ◦ g )( s ) = b for all s ∈ Y \ { c } and ( h ◦ g ) ′ ( c ) = h ′ ( g ( c )) g ′ ( c ) = h ′ ( a ) g ′ ( c ) = 0. Therefore, using h ◦ g in the definition of η Y △ ( h ( a )), we have η Y △ ( h ◦ g )( c ) ≤ η Y ( c ) | ( h ◦ g ) ′ ( c ) | .By (3.3) and using the chain rule, it follows that η Y △ ( h ◦ g )( c ) | h ′ ( a ) | ≤ η Y ( c ) | g ′ ( c ) | ≤ η Y Ω ( a ) + ǫ . Letting ǫ goes to zero, we obtain η Y △ ( h ( a )) | h ′ ( a ) | ≤ η Y Ω ( a ).This completes the proof. (cid:3) We now provide an example which demonstrate Proposition 3.9 in non-simply connecteddomains.
Example 3.12.
Let Y = D ∗ := D \ { } , the punctured unit disk and Ω = C ∗ := C \ { } ,the punctured plane. We shall prove that for all w ∈ C ∗ η D ∗ C ∗ ( w ) = η C ∗ ( w ).As stated in [9, p. 322] (see also [8, (4.1)]), the Hurwitz covering map from D onto C \ { } is obtained by the infinite product representation(3.4) g ( w ) = 16 w ∞ Y n =1 (cid:16) w n w n − (cid:17) , | w | < s ∈ D , it is well known that the map T ( z ) = ( z − s ) / (1 − sz ) defines a M¨obius trans-formation of D onto itself. Clearly, T ( s ) = 0, T ′ ( s ) = (1 + | s | ) / (1 − | s | ) >
0. Since g isa holomorphic covering map and T is a one-one holomorphic map on D , the composition g ◦ T is also a holomorphic covering map satisfying ( g ◦ T )( s ) = 0 and ( g ◦ T )( t ) = 0 for all t ∈ D \ { s } . Furthermore, ( g ◦ T ) ′ ( s ) > g ′ (0) = 16 > T ′ ( s ) >
0. By the distance decreasing property we have(3.5) η C \{ } (( g ◦ T )( s ))( g ◦ T ) ′ ( s ) = η D ( s ).Restricting the function g ◦ T onto D ∗ and plugging it in the definition of η D ∗ C \{ } (0) we obtain η D ∗ C \{ } (0) ≤ η D ∗ ( s )( g ◦ T ) ′ ( s ).Now, we choose a sequence s n ∈ D ∗ such that | s n | →
1. By using the same argument as above,we can find M¨obius transformations T n from D onto itself with T ( s n ) = 0 and T ′ ( s n ) > η D ∗ C \{ } (0) ≤ η D ∗ ( s n )( g ◦ T n ) ′ ( s n ) = η D ∗ ( s n ) η D ( s n ) η C \{ } (0),since ( g ◦ T ) ′ ( s n ) = 0. We notice from [8, Section 2] that the Hahn density of the puncturedunit disk obtained by (see [7, (2)]) S D ∗ ( s n ) = 1 + | s n | | s n | (1 − | s n | ) GENERALIZED HURWITZ METRIC 11 exceeds the Hurwitz density. Thus, we obtain η D ∗ C \{ } (0) ≤ η D ∗ ( s n ) η D ( s n ) η C \{ } (0) ≤ S D ∗ ( s n ) η D ( s n ) η C \{ } (0) = 1 + | s n | | s n | (1 − | s n | )(1 − | s n | ) η C \{ } (0).Now, letting | s n | → η D ∗ C \{ } (0) ≤ η C \{ } (0). The reverse inequality is followed byProposition 3.5. Now, by using the holomorphic functions f ( w ) = 1 − w and h ( w ) = bw (forsome complex constant b ) in the distance decreasing property for the generalized Hurwitzdensity, it follows that, both the metrics coincide on C ∗ . That is, η D ∗ C ∗ ( w ) = η C ∗ ( w ) for all w ∈ C ∗ . (cid:3) Next we define the generalized Hurwitz distance between two points in a domain.
Definition 3.13.
Let Ω ⊂ C and Y ( C be domains. For w , w ∈ Ω, define η Y Ω ( z , z ) = inf Z γ η Y Ω ( w ) | dw | ,where the infimum is taken over all rectifiable paths γ in Ω joining z to z .Proof of the following theorem is similar to that of Theorem 2.3. Theorem 3.14.
Let Y ( C be a domain. If Ω is a hyperbolic domain, then (Ω, η Y Ω ) is acomplete metric space. We do have also the distance decreasing property in the global sense whose proof followsthe steps of the proof of Proposition 2.5.
Theorem 3.15.
Let Y be a proper subdomain of C and Ω, △ be any subdomain of C . If h is a one-to-one holomorphic map from Ω to △ , then η Y △ ( h ( w , w )) ≤ η Y Ω ( w , w ), for all w , w in Ω.Note that, till now we have derived all the results of the generalized Hurwitz density η Y Ω for a base domain Y . In the next theorem we will see the comparison between generalizedHurwitz densities when the range domain is fixed while the source domain is varying. Theorem 3.16.
Let Y , Y be proper subdomains of C and Ω be any subdomain of C . Iffor every point b ∈ Y , there exists a point a ∈ Y and a holomorphic covering map h b from Y \ { a } onto Y \ { b } which extends to a holomorphic map from Y onto Y with h b ( a ) = b , h b ( w ) = b for any w ∈ Y \ { a } and h ′ b ( a ) = 0, then η Y Ω ( ζ ) ≤ η Y Ω ( ζ ), for all ζ in Ω. Proof.
Let ζ be any arbitrary point in Ω and ǫ be a positive real number. By definition of η Y Ω ,there exists a point b in Y and a holomorphic function g from Y to Ω with g ( b ) = ζ , g ( s ) = ζ for any s ∈ Y \ { b } , g ′ ( b ) = 0 such that(3.6) η Y Ω ( ζ ) ≥ η Y ( b ) | g ′ ( b ) | − ǫ .Since for every point b ∈ Y , there exists a point a in Y and a holomorphic covering h b from Y onto Y with h b ( a ) = b , h b ( w ) = b for any w ∈ Y \ { a } , g ′ ( a ) = 0, by Theorem A wehave(3.7) η Y ( b ) | h ′ b ( a ) | = η Y ( a ).Note that, the composition g ◦ h b is a holomorphic function from Y to Ω with ( g ◦ h b )( a ) = ζ , ( g ◦ h b )( w ) = ζ for any w in Y and ( g ◦ h b ) ′ ( a ) = g ′ ( b ) h ′ b ( a ) = 0. Therefore, by thedefinition of η Y Ω , we obtain(3.8) η Y Ω ( a ) ≤ η Y ( a ) | ( g ◦ h b ) ′ ( a ) | = η Y ( a ) | g ′ ( b ) h ′ b ( a ) | .By (3.6), (3.7), (3.8), it follows that η Y Ω ( ζ ) ≥ η Y ( b ) | g ′ ( b ) | − ǫ = η Y ( a ) | g ′ ( b ) || h ′ b ( a ) | − ǫ ≥ η Y Ω ( ζ ) − ǫ .Letting ǫ goes to zero, we have η Y Ω ( ζ ) ≥ η Y Ω ( ζ ), which completes the proof our result. (cid:3) We look forward for the existence of non-simply connected domains Y and Y validatingthe statement of Theorem 3.16, however, they remain open due to their non-trivial nature. Corollary 3.17. If Y ( C is a simply connected domain, then for all proper subdomains Y and Ω of C , we have η Ω ( w ) = η D Ω ( w ) = η Y Ω ( w ) ≤ η Y Ω ( w ), for all w in Ω. Proof.
Since Y ( C is a simply connected domain, by Riemann Mapping Theorem, thereexists a conformal homeomorphism f from Y onto D . Furthermore, there exists a Hurwitzcovering map T b from D onto Y for every b in Y . Thus, the composed map T b ◦ f is aholomorphic covering from Y \ { f − (0) } onto Y \ { b } , which extends to a holomorphicfunction from Y to Y with ( T b ◦ f )( f − (0)) = 0, ( T b ◦ f ) ′ ( f − (0)) = 0 and ( T b ◦ f )( s ) = b forall s in Y \ { f − (0) } . Taking h b = T b ◦ f in Theorem 3.16, we obtain the desired result. (cid:3) Two subdomains Y and Y of C are conformally equivalent if there exists a holomorphicbijection f from Y to Y . GENERALIZED HURWITZ METRIC 13
Corollary 3.18.
Let Ω ⊂ C be any arbitrary domain. If Y ( C and Y ( C are conformallyequivalent domains, then η Y Ω ( w ) = η Y Ω ( w ) for all w in Ω. 4.
Lipschitz Domain
In this section, one of our main objectives is to study the situations, in terms of theHurwitz non-Lipschitz domains, when the Hurwitz density coincides with the generalizedHurwitz density. The following notations are useful in the definition of Hurwitz Lipschitzdomains. Let Y be a hyperbolic domain and Ω be a subdomain of Y . If i is the inclusionmap from Ω to Y , the global contraction constant gl η (Ω, Y ) is defined by gl η (Ω, Y ) := sup w , w ∈ Ω, w = w η Y ( w , w ) η Ω ( w , w ).If Y is any proper subdomain of C , then the infinitesimal contraction constant is defined as l η (Ω, Y ) := sup w ∈ Ω η Y ( w ) η Ω ( w ).Since the inclusion map is an injective holomorphic function from Ω to Y , by the distancedecreasing property of Hurwitz density, we have η Y ( w ) ≤ η Ω ( w ) for every w in Ω. Further-more, by Proposition 2.5 it follows that η Y ( w , w ) ≤ η Ω ( w , w ) for all w and w in Ω.Thus, both infinitesimal and global contraction constants are less than or equal to 1. Theorem 4.1.
Let Y ( C be a domain. If Ω is a subdomain of Y , then gl η (Ω, Y ) ≤ l η (Ω, Y ) ≤ Furthermore, the inclusion map i from Ω to Y is a strict infinitesimal contrac-tion map, whenever Ω is a proper subdomain of Y . Proof.
Let w , w be any two points in Ω and γ ⊂ Ω be any path joining w and w suchthat η Ω ( w , w ) = Z γ η Ω ( w ) | dw | .By the definition of η Y ( w , w ), it follows that η Y ( w , w ) ≤ Z γ η Y ( w ) η Ω ( w ) η Ω ( w ) | dw | ≤ l η (Ω, Y ) η Ω ( w , w ).Thus, gl η (Ω, Y ) ≤ l η (Ω, Y ).The proof of the second part of our theorem follows from [8, Theorem 6.1]. (cid:3) Definition 4.2.
Let Ω be a subdomain of a domain Y in C . Then Ω is called a Hurwitz Lip-schitz subdomain of Y , if the inclusion map from Ω to Y is a strict infinitesimal contraction.That is, the infinitesimal contraction constant l Ω is strictly less than 1. By Proposition 2.5, for any proper subdomain Ω of C , we have η Y Ω ≥ η Ω . However, in thefollowing theorem, we find a condition on Y so that for every proper subdomain Ω of C , theHurwitz and the generalized Hurwitz densities coincide. We adopt the proof technique from[10, Theorem 2.1] where the author compares the hyperbolic density with the Kobayashidensity. Theorem 4.3. If Ω is any proper subdomain of C and Y is a Hurwitz non-Lipschitz subdo-main of D , then we have η Y Ω ( w ) = η Ω ( w ), for every w in Ω. Proof.
To show that η Y Ω = η Ω , we only need to show that η Y Ω ≤ η Ω , as the relation η Y Ω ≥ η Ω always holds. Since Ω ( C , for every point w ∈ Ω there exists a Hurwitz covering map g from D onto Ω with g (0) = w , g ( s ) = w for all s ∈ D \ { } and g ′ (0) = 0. We now pre-compose g with a M¨obius transformation T of D that maps s in Y to the origin. Therefore g ◦ T isa holomorphic covering of Ω from D \ { s } onto Ω \ { w } , which extends to a holomorphicfunction from D to Ω with ( g ◦ T )( s ) = w , ( g ◦ T )( t ) = w for all t in D \ { s } and ( g ◦ T ) ′ ( s ) = g ′ (0) T ′ ( s ) = 0. Thus, by Theorem A, we have(4.1) η Ω ( w ) | ( g ◦ T ) ′ ( s ) | = η D ( s ).Now, let h be the restriction of g ◦ T on Y . By the definition of η Y Ω , we obtain η Y Ω ( w ) ≤ η Y ( s ) | h ′ ( s ) | = η Y ( s ) | ( g ◦ T ) ′ ( s ) | .On the other hand, by the help of (4.1), we have η Y Ω ( w ) ≤ η Y ( s ) η D ( s ) η Ω ( w ).Since Y is a non-Lipschitz Hurwitz subdomain of D , by choosing s in Y appropriately, η Y ( s ) /η D ( s ) can be made as close to 1 as we wish. Thus, we can say that η Y Ω ( w ) ≤ η Ω ( w ).Since w ∈ Ω is an arbitrary element, therefore we have η Y Ω ( w ) = η Ω ( w ) for all w in Ω. (cid:3) In order to generalize Theorem 4.3 for a broader class of domains in C , we now discussthe notion of quasi-bounded domains as follows. Definition 4.4.
A domain Y in C is said to be quasi-bounded if the smallest simply connectedplane domain containing Y is a proper subset of C . We denote the smallest simply connecteddomain by ˆ Y . Example 4.5.
All bounded domains in C are quasi-bounded. GENERALIZED HURWITZ METRIC 15
The following theorem is an analog of [10, Theorem 2.2] from the notion of hyperbolicdensity to the notion of Hurwitz density.
Theorem 4.6. If Ω is any proper subdomain of the complex plane and Y is quasi-bounded,non-Lipschitz Hurwitz subdomain of ˆ Y , then η Y Ω ( w ) = η Ω ( w ), for all w in Ω. Proof.
Since, ˆ Y is simply connected, by Riemann Mapping Theorem there exists a conformalhomeomorphism h from ˆ Y onto D . Note that the conformal mappings are isometries for theHurwitz metric (see [8, Corollary 6.2]), and thus we have(4.2) η D ( h ( s )) | h ′ ( s ) | = η ˆ Y ( s )for all s in ˆ Y . Furthermore, the restriction of h to Y , resulting a conformal homeomorphismfrom Y onto h ( Y ). Therefore, we have(4.3) η h ( Y ) ( s ) | h ′ ( s ) | = η Y ( s ).By using (4.2), (4.3) and the definition of l η ( Y , ˆ Y ), we obtain l η ( Y , ˆ Y ) = sup z ∈ Y η Y ( z ) η ˆ Y ( z ) = sup z ∈ h ( Y ) η h ( Y ) ( z ) η D ( z ) = l η ( h ( Y ), D ).Thus, Y is a non-Lipschitz Hurwitz subdomain of ˆ Y if and only if h ( Y ) is non-LipschitzHurwitz subdomain of D . By Theorem 4.3, it follows that(4.4) η h ( Y )Ω ( w ) = η Ω ( w )for all w in Ω. Since Y and f ( Y ) are conformally homeomorphic by the map f , by Corol-lary 3.18, we have(4.5) η Y Ω ( w ) = η h ( Y )Ω ( w )for all w in Ω. Combining (4.4) and (4.5), we obtain η Y Ω ( w ) = η Ω ( w ),as desired. (cid:3) Acknowledgement.
The authors would like to thank the referees for their careful readingof the earlier versions of the manuscript and useful remarks. The research work of Arstu issupported by CSIR-UGC (Grant No: 21/06/2015(i)EU-V) and of S. K. Sahoo is supportedby NBHM, DAE (Grant No: 2 / / / NBHM (R.P.)/R & D II/13613). The authorsexpress their gratitude to Professor Toshiyuki Sugawa for useful discussion on this topic.
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