二つの双曲的リーマン面間の解析写像へのケーベの定理の一つの拡張

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(1)Title. 二つの双曲的リーマン面間の解析写像へのケーベの定理の一つの拡張. Author(s). 長田, 正幸. Citation. 北海道教育大学紀要. 第二部. A, 数学・物理学・化学・工学編, 28(2) : 61-64. Issue Date. 1978-02. URL. http://s-ir.sap.hokkyodai.ac.jp/dspace/handle/123456789/6014. Rights. Hokkaido University of Education.

(2) Journal of Hokkai do University of Education (Section II A) Vol. 28, No. 2 February 1978. ^mMH±WS (^ 2 Â ) ^ 28 ^ ^ 2 ^ Bgffi 53 ^ 2 ^. An Extension of Koebe's Theorem to an Analytic Mapping between Two Hyperbolic Riemann Surfaces. Masayuki OSADA Mathematics Laboratory, Sapporo College, Hokkaido University of Education. Sapporo 064. r.^c7)%? U —^>®^^?tjf^:ft^^^—<^^?l^—^^^1. S IE. ^m?^±^mwwm. Abstract We shall extend a well-known theorem of Koebe which is stated as follows.. Theorem of Koebe: Let f(.z) be a bounded analytic function in an open unit disk U={ \z <1}. Let {Cn}^=i be a sequence of curves intersecting two radii and tending to. 9U={\z\=l}. //Max|/(z)h0(n^^), then f{z)=0. zeCn. We shall extend this theorem to the case of an analytic mapping between two hyperbolic Riemann surfaces using the Wiener compactifications of hyperbolic Riemann surfaces.. Our main result is as follows: Let 0 be an analytic mapping of a hyperbolic Riemann surface R into a hyperbolic Riemann surface R'. Suppose {Fn}^=\ is a sequence of regvlar closed subsets of R such that H Fn=^nd lim l/?n^0. If {(/)(, Fn)}^=i converges uniformly to a n=l. nôo. point on R\ then 0 is a constant mapping. Introduction Koebe's theorem for a bounded analytic function in an open unit disk L/r={|^|<l} is a very interesting result, because it is an application of Fatou's theorem and Riesz's theorem. In this paper, we shall extend Koebe's theorem to the case of an analytic mapping of a hype-. rbolic Riemann surface into a hyperbolic Riemann surface using the Wiener compactifications of hyperbolic Riemann surfaces and the theorem of Riesz type (Satz 8.9 in [ 1 ]) for an analytic mapping of a hyperbolic Riemann surface into another Riemann surface.. (61).

(3) M. OSADA. Preliminaries We shall use the same notations and terminologies as in [ 1 ], for instance, Ip, R^y, Fatou. mapping and etc. Similarly, we shall use the same notations and terminologies as in [ 2 ], for instance, F , a), a regular closed subset and etc.. Main result We shall prove THEOREM 1. Let 0 be an analytic mapping of a hyperbolic Riemann surface R into a hyperbolic Riemann surface R'. Suppose {Fn}^=i is a sequence of regular closed subsets of R such that H Fn=â-nd lim l^Ô. If {(p{Fn)}^=i converges uniformly to a point on R\ then 0 is n=i. '. n-oo. a constant mapping.. PROOF. Suppose 0 is a nonconstant mapping. Set K =- H F^. By the Corollary to Lmma 3 n=l. in [2], Ip- „ decreases to (D(T?) and (n(70=if=0. Since R and ^? are hyperbolic, 0 is a Fatou mapping. Let 0* be a continuous extension of 0 from R"w into 7?/^." Then. ^w=^(r}Fnw] ,n=l. c0*GFT). C0*(Fj'y =~^Fn)w.. Therefore 0*(^)czH </>(Fn)w'. Since {</){Fn)}^=i converges uniformly to a point on M=l. ^. .... oo. R'', (/)(Fn')w=(/)(Fn){n^no) for some positive integer no. Hence 0*(7T)C H 0(J'n)andso M=l. (f)*[K) is a point on R'. By Satz 8.9 in [I], oo{K)=0. This is a contradiction.. COROLLARY. Let /(z) be a bounded analytic function on a hyperbolic Riemann surface R. Suppose {Fn}^=i is a sepuence of regular closed subsets of R such thst H Fn=ând limlp^Ô. n=l. n-oo. ^/limsup|/(^)|=0, then /(z)=0. n-oo zeF~n. Similarly, we can prove. THEOREM 2. Let 0 6e an analytic mapping of a hyperbolic Riemann snrface R into a hyperbolic Riemann surface R'', Let {Kn}^=i be a sequence of regular compact subsets of R such that H U Km=^ a-nd lim l/<n^0, ^{0(7^n}}^=i converges uniformly to a point on R', n=i. m=n. n-»oo. then 0 is a constant mapping. COROLLARY. Let f(z) be a bounded analytic function on a hyperbolic Riemann surface R. Let {Kn}^=i be a sepuence of regular compact subsets of R such that H U Km=^ and. Urn lKn=f=0. ^/lim Max|/(^)| =0, ^%/(^)^0. References [ 1 ] Constantinescu, C. and Cornea, A. (1963), Ideale Rander Riemannscher Flachen, Springer, Berlin-Gottingen-Heiderberg.. (62).

(4) Koebe's Theorem. [ 2] Tanaka, H. (1968), On Kuramochi's function-theoretic separative metrics on Riemann surfaces, J. Sci. Hiroshima Univ., Ser. A-I, 32, p. 309-330.. (63).

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