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Main theorem on covering surfaces. (English) Zbl 0831.30005

Let V be the Riemann sphere with unit diameter. Let F 0 be a connected domain on V, whose boundary F 0 are analytic Jordan curves {Λ i } l=1 q , where the spherical distance between any two curves is not smaller then d(0,0·5). Let L(Λ i ) be sets of all closed curves on F 0 , or all curves on F 0 , whose both end points lie on the same Λ i . Consider the two domains which are surrounded by F 0 and cL(Λ i ), and let D(F 0 ,c)F 0 be one of the domains whose area is smaller. Let

A(Λ i )=supdD(F 0 ,c) |c|,cL(Δ i ),
A(F 0 )=maxA ( Λ i ) ; i = 1 , 2 , ... , q,

where dD denotes the diameter of D, and |c| denotes the measure of c. The A(F 0 ) is called the cuted ratio of F 0 . Let F be a connected finite covering surface of F 0 . Let S=|F| |F 0 |. We say that S is the mean covering number of F 0 . Let L be the length of the relative boundary of F. We improve the Ahlfors’s inequality as following:

ρ + (F)>ρ(F 0 )S-32π 2 A(F 0 )L d 3 ,

where ρ(F) is the characteristics of F, ρ + =max{0,ρ}. If the boundary {Λ i } reduces to q single points, or circles, then ρ + (F)>ρ(F 0 )S-32π 2 L d 3 .

Reviewer: D.Sun
MSC:
30C25Covering theorems in conformal mapping theory