Definitions for quasiregular mappings

Мартио, Олли; Rickman, Seppo; Väısälä, Jussi

doi:10.5186/aasfm.1969.448

Cited by 181 publications

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“…We denote by K I (f ) the inner dilatation of a quasiregular mapping f . For the various definitions and the properties of quasiregular mappings, we refer to [10], [14], [17]. In the sequel, we assume tacitly that every quasiregular mapping that we consider is non-constant.…”

Section: Introductionmentioning

confidence: 99%

Geometric versions of Schwarz’s lemma for quasiregular mappings

Betsakos

2011

Proc. Amer. Math. Soc.

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Abstract. We prove monotonicity and distortion theorems for quasiregular mappings defined on the unit ball B n of R n . Let K I (f ) be the inner dilatation of f and let α = K I (f ) 1/(1−n) . Let m n denote n-dimensional Lebesgue measure and c n be the reduced conformal modulus in R n . We prove that the functions r −nα m n (f (rB n )) and r −α c n (f (rB n )) are increasing for 0 < r < 1. These results can be viewed as variants of the classical Schwarz lemma and as generalizations of recent results by Burckel et al. for holomorphic functions in the unit disk.

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Section: Introductionmentioning

confidence: 99%

Geometric versions of Schwarz’s lemma for quasiregular mappings

Betsakos

2011

Proc. Amer. Math. Soc.

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“…In the case of quasiregular mappings K I -inequality with the constant C I (G) ≡ 1 was first proved by Martio, Rickman and Väisälä [10]. Later Martio improved this result by improving the constant C I (G), see [9].…”

Section: N−1 Locmentioning

confidence: 81%

“…Moreover, inequalities similar to (1.1) has been applied widely to study properties of quasiregular mappings, see e.g. [9,8,10].…”

Section: N−1 Locmentioning

confidence: 99%

Sharpness of the differentiability almost everywhere and capacitary estimates for Sobolev mappings

Hencl¹,

Tengvall²

2017

Rev. Mat. Iberoam.

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We give sharp conformal conditions for the differentiability in the Sobolev space W_{\mathrm {loc}}^{1,n-1}(\Omega, \mathbb R^{n}) . Furthermore, we show that the space W_{\mathrm {loc}}^{1,n-1}(\Omega, \mathbb R^{n}) can be considered as the borderline space for some capacitary inequalities.

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“…for an arbitrary Borel set A in the domain D such that N.f; A/ < 1 and an arbitrary family of curves in A (see Theorem 3.2 in [6] or Theorem 6.7 in [4], Chap. II).…”

Section: Introductionmentioning

confidence: 99%

On the openness and discreteness of mappings with unbounded characteristic of quasiconformality

Sevost’yanov

2012

Ukr Math J

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The paper is devoted to the investigation of topological properties of space mappings. It is shown that orientation-preserving mappings f W D ! R n in a domain D R n ; n 2; which are more general than mappings with bounded distortion, are open and discrete if a function Q corresponding to the control of the distortion of families of curves under these mappings has slow growth in the domain f .D/; e.g., if Q has finite mean oscillation at an arbitrary point y 0 2 f .D/:

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Definitions for quasiregular mappings

Cited by 181 publications

References 0 publications

Geometric versions of Schwarz’s lemma for quasiregular mappings

Geometric versions of Schwarz’s lemma for quasiregular mappings

Sharpness of the differentiability almost everywhere and capacitary estimates for Sobolev mappings

On the openness and discreteness of mappings with unbounded characteristic of quasiconformality

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