Carathéodory density of the Hurwitz metric on plane domains
aa r X i v : . [ m a t h . C V ] A p r CARATH ´EODORY DENSITY OF THE HURWITZ METRIC ON PLANEDOMAINS
ARSTU † AND SWADESH KUMAR SAHOO † Abstract.
It is well-known that the Carath´eodory metric is a natural generalization ofthe Poincar´e metric, namely, the hyperbolic metric of the unit disk. In 2016, the Hurwitzmetric was introduced by D. Minda in arbitrary proper subdomains of the complex planeand he proved that this metric coincides with the hyperbolic metric when the domains aresimply connected. In this paper, we define a new metric which generalizes the Hurwitzmetric in the sense of Carath´eodory. Our main focus is to study its various basic propertiesin connection with the Hurwitz metric. Introduction
Studying families of holomorphic functions associated with the hyperbolic metric alwaysremains the hot topic in geometric function theory. Several researchers (for instance see[2, 5, 6]) have introduced new metrics which are closely related to the hyperbolic metric andestablished their comparison properties in possible situations. In particular, the generalizedKobayashi metric is one such metric which is always greater than or equal to the hyperbolicmetric (see [3, Proposition 1]). The generalized Kobayashi metric in a domain is definedby the smallest push forward of the hyperbolic metric from a hyperbolic domain to a planedomain by holomorphic functions. In [3], it is further proved that the generalized Kobayashimetric agrees with the hyperbolic metric on simply connected domains (see also [4]). Coin-cidence of the hyperbolic and the generalized Kobayashi densities on other plane domainsare studied in [7]. Moreover, the Kobayashi density satisfies the generalized Schwarz lemmafor holomorphic function between two domains.In 2016, Minda introduced a new metric, namely, the Hurwitz metric that also exceeds thehyperbolic metric in hyperbolic domains (see [6]) and investigated several basic propertiessuch as distance decreasing property, conformal invariance property, domain monotonicityproperty, bilipschitz equivalent properties with the hyperbolic and the quasihyperbolic met-rics. In our recent work (see [1]), we studied a new metric that generalizes the Hurwitzmetric in the sense of Kobayashi. This new work focuses on some basic properties of thisgeneralized metric.
Mathematics Subject Classification.
Primary 30F45; Secondary 30C20, 30C80.
Key words and phrases. hyperbolic density, Hurwitz density, Kobayashi density, Carath´eodory density,conformal mapping, covering mapping.
On the other hand, the classical Carath´eodory metric is another generalized metric whichis always less than or equal to the hyperbolic metric. The Carath´eodory metric in a domainis the largest pull back of the hyperbolic metric. Similar to the case of the Kobayashimetric, the Carath´eodory metric also agrees with the hyperbolic metric on simply connecteddomains. Furthermore, it satisfies the generlized Schwarz lemma for holomorphic functionbetween two domains. Analogous to the Carath´odory metric, in this paper, we generalizethe Hurwitz metric and study its basic properties.Rest of this document is organized as follows: Section 2 contains preliminary informationincluding terminology, definitions and well known results. We define the generalized Hurwitzmetric in the sense of Carath´eodory Section 3 and study its various properties includingdistance decreasing property for special class of holomorphic function between two domains.Finally, Section 4 is devoted to the distance between two points induced by the generalizedHurwitz metric. 2. preliminaries
Throughout the paper, unless it is specified, we assume that Ω is an arbitrary domainand Y is a proper subdomain in C , the complex plane. Symbolically, we write Ω ⊂ C and Y ( C . We denote H (Ω, Y ) by the set of all holomorphic functions from Ω into Y . For afixed w ∈ Ω, we define the following notation: H sw (Ω, Y ) = { h ∈ H (Ω, Y ), h ( w ) = s , h ( z ) = s for all z ∈ Ω \ { w }} .The open unit disk { z ∈ C : | z | < } is denoted by D . The family H w ( D , Y ) is known asthe Hurwitz family . More about the Hurwitz family and several other classes of holomorphicfunctions analogous to the Hurwitz family are discussed in [6]. By setting F ′ (0) = r Y ( w ) = max { h ′ (0) : h ∈ H w ( D , Y ) } ,the Hurwitz density is defined as η Y ( w ) = 2 F ′ (0) = 2 r Y ( w ).An equivalent definition of the Hurwitz density can be found in [1]. A domain Y is said tobe hyperbolic provided its complement C \ Y contains at least two points. The supremumof {| h ′ (0) | : h ∈ H ( D , Y ) } leads to the definition of the hyperbolic density. Indeed, for anypoint w ∈ Y the hyperbolic density λ Y is defined as λ Y ( w ) = 2 | g ′ (0) | ,where g is a universal covering mapping from D onto Y . Note that existence of such g isguaranteed by the uniformization theorem. Analogous to the hyperbolic density, we nowdescribe the maximizer for the Hurwitz family. For s ∈ Y , there exists a holomorphic ARATH´EODORY DENSITY OF THE HURWITZ METRIC 3 covering map F : D \ { } → Y \ { s } which extends to D holomorphically in such a waythat F (0) = s and F ′ (0) >
0. This is determined by the subgroup generated by the curve,namely the circle centered at s and radius ρ , for some ρ >
0, in Y \ { s } of the fundamentalgroup of Y \ { s } . The function F is the unique extremal function for the Hurwitz-extremalproblem max { h ′ (0) : h ∈ H w ( D , Y ), h ′ (0) > } and is defined as the Hurwitz covering map(see [6]). 3.
Carath´eodory density of the Hurwitz metric
In [1], by adopting the idea of the Kobayashi metric, we generalized the Hurwitz densityfor a domain Ω ⊂ C and Y ( C as follows: 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 h ∈ H ( Y , Ω)satisfying h ( s ) = w , h ( t ) = w for all t ∈ Y \ { s } , and h ′ ( s ) = 0. We name the quantity η Y Ω by the Kobayashi density of the Hurwitz metric of Ω relative to Y .As stated at the end of Section 1, this section is devoted to the introduction of a newdensity that generalizes the Hurwitz density in the sense of Carath´eodory. This is definedas follows: Definition 3.1.
Let w ∈ Ω ( C . For an element s ∈ D , we define a new quantity(3.1) C D , s Ω ( w ) = sup η D ( h ( w )) | h ′ ( w ) | ,where the supremum is taken over all h ∈ H (Ω, D ) such that h ( w ) = s , h ( z ) = s for all z ∈ Ω \ { w } , i.e. for all h ∈ H sw (Ω, D ). We call this quantity by the Carath´eodory density ofthe Hurwitz metric of Ω relative to D . Setting C D Ω := C D ,0Ω . Remark 3.2. (1) Note that on simply connected domains the Hurwitz density agreeswith the hyperbolic density, so one can replace η D by the hyperbolic density λ D inDefinition 3.1.(2) If Ω = C , then by Liouville’s theorem, the only holomorphic function from Ω into D is a constant function, which does not belong to the class H sw (Ω, D ). Hence, it canbe defined that C D , s C ( w ) = 0 when the set H sw (Ω, D ) becomes empty. It suggests usto assume that H sw (Ω, Y ) = ∅ throughout the paper for an arbitrary base domain Y ( C .The first basic property of the Carath´eodory density of the Hurwitz metric C D , s Ω is thatthe supremum is attained by some holomorphic function h ∈ H sw (Ω, D ) in (3.1). ARSTU AND S. K. SAHOO
Proposition 3.3.
Let Ω ( C be a domain and H w (Ω, D ) = ∅ . Then, the Carath´eodorydensity of the Hurwitz metric C D Ω can be computed by the formula: C D Ω ( w ) = 2 max {| h ′ ( w ) | : h ∈ H w (Ω, D ) } . Proof.
Since the members of the family H w (Ω, D ) are uniformly bounded by 1, by Mon-tel’s Theorem, H w (Ω, D ) is a normal family. By Definition 3.1 there exists a sequence ofholomorphic functions h n ∈ H w (Ω, D ) such that 2 | h ′ n ( w ) | → C D Ω ( w ), since h n ( w ) = 0 and η D (0) = λ D (0) = 2. Furthermore, by the open mapping theorem, there exists a subsequence h n k of h n which converges to either an open map h or a constant map. Since h n ∈ H (Ω, D ),it follows that | h ( z ) | ≤ z ∈ Ω. Note that, if h ( z ) attains 1 for some z ∈ Ω, then bythe maximum modulus principle, | h | = 1, contradicting to the fact that h ( w ) = 0. Moreover,by Hurwitz Theorem, there exists an N ∈ N such that h n k and h have the same number ofzeros for all n k ≥ N in some neighborhood of w . Since h ( z ) = 0 for all z ∈ Ω \ { w } , weconclude by the uniqueness of limit that 2 | h ′ ( w ) | = C D Ω ( w ), which completes the proof. (cid:3) Remark 3.4.
By a suitable composition of the disk automorphism with the function ob-tained in Proposition 3.3, we can prove the existence of the holomorphic function h inDefinition 3.1 when s = 0.Alike to the case of coinciding of the hyperbolic and Carath´eodory density on simplyconnected domains, we now prove that the Hurwitz density η Ω and the Carath´eodory densityof the Hurwitz metric C D , s Ω too agree on simply connected domains Ω. Proposition 3.5. If Ω ( C is a simply connected domain, then the Carath´eodory density ofthe Hurwitz metric C D , s Ω coincides with the Hurwitz density η Ω as well as with the Kobayashidensity of the Hurwitz metric η D Ω . That is, we have C D , s Ω ≡ η Ω ≡ η D Ω . Proof.
By the distance decreasing property of the Hurwitz density (see [6, Theorem 6.1]),for a point w ∈ Ω and for any h ∈ H sw (Ω, D ) we have η D ( h ( w )) | h ′ ( w ) | ≤ η Ω ( w ). By takingsupremum over all h ∈ H sw (Ω, D ), in one hand, we obtain C D , s Ω ( w ) ≤ η Ω ( w ). On the otherhand, to prove the reverse inequality, we consider the conformal homeomorphism f : Ω → D which is guaranteed by Riemann Mapping Theorem. By [6, Corollary 6.2], it follows that η Ω ( w ) = η D ( h ( w )) | h ′ ( w ) | ≤ C D , s Ω ( w ),where the inequality holds by Definition 3.1. Thus, we have the identity C D , s Ω ≡ η Ω .The second required identity follows from [1, Corollary 3.10], completing the proof. (cid:3) Due to [1, Corollary 3.10], the Kobayashi density of the Hurwitz metric η D Ω and the Hurwitzdensity η Ω both agree on any domain Ω, whereas, in the following result we show that on non-simply connected domains the Carath´eodory density of the Hurwitz metric C D , s Ω is strictlyless than the Hurwitz density η Ω . ARATH´EODORY DENSITY OF THE HURWITZ METRIC 5
Proposition 3.6.
Let Ω ( C be a non-simply connected domain and C D Ω > . Then for anelement w ∈ Ω we have the strict inequality: C D Ω ( w ) < η Ω ( w ). Proof.
Let w ∈ Ω. Since Ω ( C , there exists a Hurwitz covering map g : D → Ω with g (0) = w . By Proposition 3.3, there exists a function h ∈ H w (Ω, D ) such that(3.2) C D Ω ( w ) = 2 | h ′ ( w ) | = η D ( h ( w )) | h ′ ( w ) | holds, since h ( w ) = 0 and η D (0) = λ D (0) = 2. Thus, we observe that the composition h ◦ g is a holomorphic function from D to D that fixes the origin. Since Ω is non-simplyconnected, the covering map g can not be one-one and hence the composition h ◦ g can neverbe conformal. Thus, by the classical Schwarz lemma we conclude the strict inequality λ D (( h ◦ g )(0)) | ( h ◦ g ) ′ (0) | < λ D (0).Note that the hyperbolic density coincides with the Hurwitz density on simply connectedhyperbolic domains (see [6, p. 15]). Therefore, it follows that(3.3) η D (( h ◦ g )(0)) | ( h ◦ g ) ′ (0) | < η D (0).Since g is a Hurwitz covering map, by [6, Theorem 6.1], we have the equality(3.4) η Ω ( g (0)) | g ′ (0) | = η D (0).Combining (3.3) and (3.4), we obtain from (3.2) that C D Ω ( w ) = η D ( h ( w )) | h ′ ( w ) | = η D (( h ◦ g )(0)) | ( h ◦ g ) ′ (0) || g ′ (0) | < η D (0) | g ′ (0) | = η Ω ( w ),where the second equality follows by the chain rule. (cid:3) Since the Hurwitz density can be defined on a proper subdomain of the complex plane, anatural way of further generalizing the Carath´eodory density of the Hurwitz metric C D , s Ω bychanging the base domain from the unit disk to a proper subdomain Y of C . The definitionis as follows: Definition 3.7.
Let Y ( C and Ω ⊂ C be domains. For w ∈ Ω and s ∈ Y , the Carath´eodorydensity of the Hurwitz metric of Ω relative to the base domain Y is defined as C Y , s Ω ( w ) = sup η Y ( h ( w )) | h ′ ( w ) | ,where the supremum is taken over all h ∈ H (Ω, Y ) such that h ( w ) = s , h ( z ) = s for all z ∈ Ω \ { w } , i.e. for all h ∈ H sw (Ω, Y ).In [1] we have noticed that the Kobayashi density of the Hurwitz metric η Y Ω exceeds overthe Hurwitz density η Ω whereas in case of the Carath´eodory density of the Hurwitz metric C Y , s Ω , we prove that it lacks the Hurwitz density on proper subdomains of C . ARSTU AND S. K. SAHOO
Proposition 3.8.
Let Ω and Y be proper subdomains of the complex plane C . If for anelement s ∈ Y , we assume C Y , s Ω > then η Ω ( w ) ≥ C Y , s Ω ( w ) holds for every w ∈ Ω .Proof. By the distance decreasing property of the Hurwitz density, for w ∈ Ω, s ∈ Y and forany h ∈ H sw (Ω, Y ) we have η Y ( h ( w )) | h ′ ( w ) | ≤ η Ω ( w ).Taking the supremum over all h ∈ H sw (Ω, Y ) on both sides, we obtain C Y , s Ω ( w ) ≤ η Ω ( w ).Since w ∈ Ω was arbitrary, we conclude the proof as desired. (cid:3)
Recall that the Hurwitz density and the hyperbolic density agree on simply connected do-mains. Analogous to this, we now prove that upon some specific conditions the Carath´eodorydensity of the Hurwitz metric C Y , s Ω and the Hurwitz density η Ω coincide and in a more specialsituation, they also coincide with the Kobayashi density of the Hurwitz metric η Y Ω . Proposition 3.9.
Let Ω, Y ( C be domains. Suppose that for every s ∈ Y there exists apoint w ∈ Ω and a holomorphic covering map g s : Ω \ { w } → Y \ { s } which extends to aholomorphic function g : Ω → Y with g ( w ) = s and g ′ ( w ) = 0 . If C Y , s Ω > , then C Y , s Ω ≡ η Ω . In particular, when Y = Ω , we have C Ω, w Ω ≡ η Ω ≡ ζ ΩΩ . Proof.
By the distance decreasing property of the Hurwitz density (see the second part of[6, Theorem 6.1]), we have η Y ( g ( w )) | g ′ ( w ) | = η Ω ( w ).Now, plugging the holomorphic covering map g s into Definition 3.7, in one hand we obtain C Y , s Ω ( w ) ≥ η Y ( g ( w )) | g ′ ( w ) | = η Ω ( w ).On the other hand, the reverse inequality follows from Proposition 3.8. Since w is arbitrary,the Carath´eodory density of the Hurwitz metric C Y , s Ω and the Hurwitz density η Ω both agreeover Ω.The proof of the second part is a combination of the above identity that we just provedand the identity proved in [1, Proposition 3.9]. (cid:3) An instant corollary to Proposition 3.9 is that on simply connected domains both theHurwitz density η Ω and the Carath’eodory density of the Hurwitz metric C Y , s Ω agree. ARATH´EODORY DENSITY OF THE HURWITZ METRIC 7
Corollary 3.10. If Ω ( C be a simply connected domain and Y ( C be an arbitrary domain,then C Y , s Ω ≡ η Ω , where s ∈ Y .Proof. Since Y ( C , there exists a Hurwitz covering map g : D → Y . Now, Ω ( C beinga simply connected domain, by Riemann Mapping Theorem, we would get a conformalmapping h : Ω → D with h ( w ) = 0 and h ′ ( w ) > w ∈ Ω. Then the composition g ◦ h is a holomorphic covering map from Ω \ { w } onto Y \ { s } for some s ∈ Y that can beextended from Ω onto Y by taking w to s with its derivative non-zero at the point w . Theproof now follows by Proposition 3.9. (cid:3) Recall that the hyperbolic density λ Ω , the Hurwitz density η Ω and the Kobayashi densityof the Hurwitz metric η Y Ω satisfy the distance decreasing property. Note that, in case of thehyperbolic metric the distance decreasing property is also known as the generalized Schwarz-Pick lemma. Alike to these properties we here show that the Carath´eodory density of theHurwitz metric C Y , s Ω too satisfies the distance decreasing property. Theorem 3.11. ( Distance decreasing property ) Let Ω , Ω ⊂ C and Y ( C be domains.If there exists a holomorphic function f from Ω into Ω with f ( a ) = b , f ( s ) = b for all s ∈ Ω \ { a } , then C Y , c Ω ( f ( a )) | f ′ ( a ) | ≤ C Y , c Ω ( a ), where c ∈ Y .Proof. If H cb (Ω , Y ) = ∅ , then C Y , c Ω = 0 and hence there is nothing to prove. Therefore,without loss of generality we assume that H cb (Ω , Y ) = ∅ .By the definition of C Y , c Ω ( b ), for every ǫ > h fromΩ into Y with h ( b ) = c , h ( s ) = c for all s ∈ Ω \ { b } for some c ∈ Y , such that(3.5) C Y , c Ω ( b ) − ǫ ≤ η Y ( h ( b )) | h ′ ( b ) | .Suppose that f is a holomorphic function from Ω into Ω with f ( a ) = b , f ( s ) = b for all s ∈ Ω \ { a } . Now the composition function h ◦ f ∈ H (Ω , Y ) satisfies ( h ◦ f )( a ) = c .Furthermore, ( h ◦ f )( t ) = c for all t ∈ Ω \ { a } as b / ∈ f (Ω ) \ { a } and c / ∈ h (Ω ) \ { b } . Now,by plugging the map h ◦ f into the definition of C Y , c Ω ( a ), it follows that(3.6) C Y , c Ω ( a ) ≥ η Y (( h ◦ f )( a )) | ( h ◦ f ) ′ ( a ) | = η Y ( h ( b )) | h ′ ( b ) || f ′ ( a ) | .Combining (3.5) and (3.6), we obtain C Y , c Ω ( a ) ≥ ( C Y , c Ω ( b ) − ǫ ) | f ′ ( a ) | which holds for every ǫ >
0. Letting ǫ →
0, we have the desired inequality. (cid:3)
ARSTU AND S. K. SAHOO
As a direct consequence of Theorem 3.11, we obtain the conformal invariance property andmonotonicity property of the Carath´eodory density of the Hurwitz metric C Y , s Ω as follows: Corollary 3.12. ( Conformal invariance property ) If f is a conformal mapping from a do-main Ω ⊂ C onto another domain Ω ⊂ C , then for a base domain Y ( C we have C Y , s Ω ( f ( w )) | f ′ ( w ) | = C Y , s Ω ( w ), for all w ∈ Ω and s ∈ Y . Corollary 3.13. ( Domain monotonicity property ) If Ω ( Ω and Y are domains as inTheorem , then C Y , s Ω ( w ) ≤ C Y , s Ω ( w ) for all w ∈ Ω and s ∈ Y . Until now, we studied the properties of the Carath´eodory density of the Hurwitz metric C Y , s Ω by fixing the base domain Y . For two different base domains, the comparison result isgiven below. Theorem 3.14.
Let Y , Y ( C and Ω ⊂ C be subdomains. If for every point b ∈ Y thereexists a point a ∈ Y and a holomorphic covering map g b : Y \ { a } → Y \ { b } which extendsto the holomorphic function with g b ( a ) = b and g ′ b ( a ) = 0 , then C Y , a Ω ( w ) ≤ C Y , b Ω ( w ) for all w ∈ Ω .Proof. By the distance decreasing property for Hurwitz density, it follows that(3.7) η Y ( g b ( a )) | g ′ b ( a ) | = η Y ( a )since g b is the extended holomorphic covering map from Y onto Y . Let ǫ > H aw (Ω, Y ) = ∅ , then C Y , a Ω = 0 and hence there is nothing to prove. Therefore, withoutloss of generality we assume that H aw (Ω, Y ) = ∅ .By the definition of C Y , a Ω , for a ∈ Y and w ∈ Ω, there exists a function h ∈ H aw (Ω, Y )such that(3.8) C Y , a Ω ( w ) ≤ η Y ( h ( w )) | h ′ ( w ) | + ǫ .Now we notice that the composed function g b ◦ h ∈ H (Ω, Y ) satisfies ( g b ◦ h )( w ) = b ,( g b ◦ h )( z ) = b for all z ∈ Ω \ { w } . Hence, g b ◦ h ∈ H bw (Ω, Y ). Applying g b ◦ h in thedefinition of C Y , b Ω ( w ), we conclude that(3.9) C Y , b Ω ( w ) ≥ η Y (( g b ◦ h )( w )) | ( g b ◦ h ) ′ ( w ) | = η Y ( g b ( a )) | g ′ b ( a ) || h ′ ( w ) | for all w ∈ Ω. Combining (3.7), (3.8), (3.9) and applying the chain rule, we obtain C Y , a Ω ( w ) ≤ η Y ( a ) | h ′ ( w ) | + ǫ = η Y ( g b ( a )) | g ′ b ( a ) || h ′ ( w ) | + ǫ ≤ C Y , b Ω ( w ) + ǫ ARATH´EODORY DENSITY OF THE HURWITZ METRIC 9 for all w ∈ Ω. Since ǫ is arbitrary, we can let it approach to zero to obtain the desiredinequality. (cid:3) Corollary 3.15. If Y and Y are conformally equivalent proper subdomains of C and Ω isan arbitrary subdomain of C , then C Y , a Ω ( w ) = C Y , b Ω ( w ) holds for every w ∈ Ω and for some a ∈ Y , b ∈ Y .Proof. We consider the inverse image of the conformal mapping in Theorem 3.14 to obtainthe reverse inequality C Y , a Ω ( w ) ≥ C Y , b Ω ( w ). (cid:3) A distance function
In this section, we consider the usual distance function associated with the Carath´eodorydensity of the Hurwitz metric C Y , s Ω for the domains Y ( C and Ω ⊂ C . Definition 4.1.
Let Y ( C and Ω ⊂ C be domains. For w , w ∈ Ω and s ∈ Y define C Y Ω ( w , w ) = inf Z γ C Y , s Ω ( w ) | dw | ,where the infimum is taken over all rectifiable paths γ ⊂ Ω joining w to w . If C Y Ω definesa metric, then we say (Ω, C Y Ω ) a metric space.It is easy to see from Definition 4.1 that C Y Ω ( w , w ) = 0 and C Y Ω ( w , w ) = C Y Ω ( w , w )for any w , w ∈ Ω. Further, it can also be verified that C Y Ω satisfies the triangle inequality.Hence, at least we can say that C Y Ω is a pseudo-metric. At present we do not know whether C Y Ω defines a metric or not. However, we have a partial solution to this whenever Ω ⊂ Y . Theorem 4.2. If Ω ⊂ Y ( C are domains, then ( C Y Ω , Ω) becomes a metric space.Proof. Since C Y Ω is a pseudo-metric on Ω, it is enough to show that C Y Ω ( w , w ) > w , w ∈ Ω. Let γ be an arbitrary rectifiable curve joining w to w in Ω.Since Ω ⊂ Y , plugging the inclusion mapping i ∈ H ww (Ω, Y ) into the definition of C Y , w Ω ( w ),we conclude that Z γ C Y , w Ω ( w ) | dw | ≥ Z γ η Y ( i ( w )) | i ′ ( w ) || dw | = Z γ η Y ( w ) | dw | By the definition of Hurwitz distance (see [1]) between two points, it follows that Z γ C Y , w Ω ( w ) | dw | > η Y ( w , w ).Now, taking infimum over γ , we obtain C Y Ω ( w , w ) ≥ η Y ( w , w ) > where the last inequality follows from [1, Theorem 2.3]. Hence (Ω, C Y Ω ) defines a metricspace, completing the proof. (cid:3) Acknowledgement.
The authors would like to thank the referee for his/her careful readingof the manuscript and useful remarks. The research work of Arstu is supported by CSIR-UGC (Grant No: 21/06/2015(i)EU-V) and of S. K. Sahoo is partially supported by NBHM,DAE (Grant No: 2 / / / NBHM (R.P.)/R & D II/13613).
References [1] Arstu and S. K. Sahoo,
A generalized Hurwitz metric , Bull. Korean Math. Soc., To appear.[2] D. A. Herron, Z. Ibragimov, and D. Minda,
Geodesics and curvature of M¨obius invariant metrics , RockyMountain J. Math., (2008), no. 3, 891–921.[3] L. Keen and N. Lakic, A generalized hyperbolic metric for plane domains , Contemp. Math., (2007),107–118.[4] L. Keen and N. Lakic,
Hyperbolic geometry from a local viewpoint , Cambridge University Press, Cam-bridge, 2007.[5] D. Minda,
The Hahn metric on Riemann surfaces , Kodai Math. J., (1) (1983), 57–69.[6] D. Minda, The Hurwitz metric , Complex Anal. Oper. Theory, (1) (2016), 13–27.[7] K. Tavakoli, Coincidence of hyperbolic and generalized Kobayashi densities on plane domains , Ann.Acad. Sci. Fenn. Math., (2009), 347–351. † Discipline of Mathematics, Indian Institute of Technology Indore, Simrol, KhandwaRoad, Indore 453 552, India
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