Download Analytic Functions Błażejewko 1982: Proceedings of a by Boris Nikolaevič Apanasov (auth.), Julian Ławrynowicz (eds.) PDF

By Boris Nikolaevič Apanasov (auth.), Julian Ławrynowicz (eds.)

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Additional info for Analytic Functions Błażejewko 1982: Proceedings of a Conference held in Błażejewko, Poland, August 19–27, 1982

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Theorem 3. ~ k(·,·) ~ !! 2 of class So(')£(Do;ncU:U», where that, for !!!!! lit P+CD). lit dense subset of !! hyperbolic simply connected domain!!! U€U, ~ ~-valued DxD D and let 0:. 2. D. P3 (Do)' Then there exists ~ 28 Jacob Burbea Fix a unit vector ~. 5) UEU and define = (S0 (z)u,u)U s(u)(Z) o It follows, since ZED. 6) Let and that o o he a Riemann mapping of ~:Q+~ ~os~u). ' Therefore, t~U)(Z)E~ L (u)

P-module and p-capaci ty in the case of open sets n Dc R Now, let us precise the concepts contained in this paper. Let X be a family of continua y and F(X) the class of admissi- ble functions p character! zed by the following conditions: p ~ 0 in n R is Borel measurable and JpdH 1 i1; 1, Vy E r (V means "for every") , y where H' is the Hausdorff linear measure. Then the p-module of X is given as inf JpPdm, pEF (X) where dm is the volume element with respect to the Lebesgue n-dimen- sional measure and the integration is taken over the whole space If F(X) 0, particular case Rn.

X)] x+1; y ~ XED k (m are said to be = 1 ,2, ... ) such that Then (1) holds. Using the notation of Lemma 3, clearly, the arc families Dk k n Ei of E ,E ~ two sets such that o 1 D with Dk n Ei 'I ~ (i = ~ 0, where the infimum is taken over all Dk~ 1,2, ••• ) Em m THEOREM 1. (11) if of all continua in to contain the complement of ~ ball then m y; 1, wn B ! J...! 1 (c) of Vaisala's paper [22J, we have Mpr E M rk k P +E M r m O P m +E M r q 1 P q' where r and r O m =r 1 =¢ q yields M r O =Mp r q1 =0 pm (m, q = 1 ,2, ••• ) • Then, taking into account Lemma 3, it follows that we may suppose, without loss of generality, that D itself is a domain, since otherwise, we can establish (11) for each component of D, separately.

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