New PDF release: Quasiconformal Mappings in the Plane:: Parametrical Methods

By Julian Lawrynowicz

ISBN-10: 3540119892

ISBN-13: 9783540119890

ISBN-10: 3540394648

ISBN-13: 9783540394648

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Extra info for Quasiconformal Mappings in the Plane:: Parametrical Methods

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17) lim sup{IISIlp/(p_ - I) 2} = c I < + ~ , p--~1+ lim sup(IIsI~/p 2)~ p-~ =c 2<+~. r o o f. As we have already remarked in Step E of the preceding proof, continuity of IIsI~ is a direct consequence of Lemma 7. 1), IISIIpl~ IIsII~-tlIsI~t, where t = (½ -I/Pi)/( ½ - I/P2) , P2>Pl >2" Since t < 1 and, by Lemma 8, IIsII2 = I, then IISIIp is strictly increasing for 2 5 p < + ~ and grows to +~ as p ~ + ~. 17) follow directly from the inequality IIsI~ ( ½ ~ A2), ~ derived in Step• E of the preceding proof, and from the formula ~ = A P ' = [(I - I/p)- 2 / p - I ] -~, where p ~ 2 and I/p+ I/p'=I, obtained in Section 5.

A slight modification of the preceding argument or a direct application of Theorem 4 together with Lemma 3 yields the following alternative formulation of this theorem: COROLLARY 5. 2) holds. We conclude this chapter by a survey of other characterizations of the class of Q-qc mappings between two fixed plane domains D'. In this survey we essentially follow Gehring [2]. Let E be a configuration consisting of a domain D and D bounded by m Jordan curves, together with n boundary points and p interior points distinguished.

The closed plane • of Theorem 3 reD" provided D" is slit along a collec- 48 I. Basic concepts and theorems tion of parallel straight-line segments which may reduce to one point sets (cf. g. Golusin [2], pp. 178-183). Now we turn our attention to the particular case n = 2. A func~r,R tion f is said to be of the class ~Q if it maps A r onto A R Qquasioonformally with f(1) = I; inn the degenerate case r = 0 we assume, i_~n addition, that f(0) = 0. Thus S~'0~ = SQ. We will see later ~r,R is nonempty iff r Q < R < r I/Q (we (Corollary 16) that the class ~Q shall always assume that r < I).

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Quasiconformal Mappings in the Plane:: Parametrical Methods by Julian Lawrynowicz

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