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• W2963907578 abstract "We give the lower bound for the modulus of the radial derivatives and Jacobian of harmonic injective mappings from the unit ball onto convex domain in plane and space. As an application we show co-Lipschitz property of some classes of qch mappings. We also review related results in planar case using some novelty. Throughout the paper we denote by , G and D open subset of R n , n> 1. Let B n (x; r) =fz2 R n :jz xj < rg; S n 1 (x; r) = @B n (x; r) (abbreviated S(x; r)) and let B n ; S = S n 1 stand for the unit ball and the unit sphere in R n , respectively. In particular, by D we denote the unit disc B 2 and T =@D we denote the unit circle S 1 in the complex plane. For a domain D in R n with non-empty boundary, we define the distance function d = dD = dist(D) by d(x) = d(x;@D) = dist(D)(x) = inffjx yj : y2 @Dg; and if f maps D onto D 0 R n , in some settings it is convenient to use short notation d = d (x) = d f (x) for d( f (x);@D 0 ). It is clear that d(x) = dist(x; D c ), where D c is the complement of D in R n . Let G be an open set in R n . A mapping f : G! R m is di erentiable at x2 G if there is a linear mapping f 0 (x) : R n ! R n , called the derivative of f at x, such that f (x + h) f (x) = f 0 (x)h +jhj(x; h) where(x; h)! 0 as h! 0. For a vector-valued function f : G! R n , where G R n , is a domain, we define j f 0 (x)j = max jhj=1 j f 0 (x)hj and l( f 0 (x)) = min jhj=1 j f 0 (x)hj;" @default.
• W2963907578 created "2019-07-30" @default.
• W2963907578 creator A5036322295 @default.
• W2963907578 date "2015-01-01" @default.
• W2963907578 modified "2023-09-23" @default.
• W2963907578 title "The Lower Bound for the Modulus of the Derivatives and Jacobian of Harmonic Injective Mappings" @default.
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• W2963907578 doi "https://doi.org/10.13140/rg.2.1.4713.9364" @default.
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