2016
DOI: 10.1007/s00605-016-0904-2
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Radii of covering disks for locally univalent harmonic mappings

Abstract: For a univalent smooth mapping f of the unit disk D of complex plane onto the manifold f (D), let d f (z 0 ) be the radius of the largest univalent disk on the manifold f (D) centered at f (z 0 ) (|z 0 | < 1). The main aim of the present article is to investigate how the radius d h (z 0 ) varies when the analytic function h is replaced by a sense-preserving harmonic function f = h + g. The main result includes sharp upper and lower bounds for the quotient d f (z 0 )/d h (z 0 ), especially, for a family of loca… Show more

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Cited by 4 publications
(3 citation statements)
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References 19 publications
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“…is proved in [13] for the orderα of the affine hull AL of any linear invariant family L. Therefore, |A 2 (ρ, ε)| ≤ α(ρ) + 1. Here α(ρ) denotes order of the harmonic function f ρ (z).…”
Section: S[h](z)| ≤ 2p(|z|) In Dmentioning
confidence: 85%
“…is proved in [13] for the orderα of the affine hull AL of any linear invariant family L. Therefore, |A 2 (ρ, ε)| ≤ α(ρ) + 1. Here α(ρ) denotes order of the harmonic function f ρ (z).…”
Section: S[h](z)| ≤ 2p(|z|) In Dmentioning
confidence: 85%
“…If B > 0 is finite, then we call B the Bloch type constant of the set P. One of the long standing open problems of determining the precise value of Bloch type constant of holomorphic mappings with one variable has attracted much attention (see [3,4,20,36,42]). For holomorphic mappings of several complex variables, the Bloch type constant does not exist unless one considers the class of functions under certain constraints.…”
Section: Denote By Ph (K) the Set Of All Pluriharmonic Mappingsmentioning
confidence: 99%
“…In the following, for f = h + g ∈ PH (k), we will use Theorems 2.2 and 2.3 to investigate the ratio B f /B h and give a sharp estimate. For the related studies of the planar harmonic mappings, see [8,10,20].…”
Section: Denote By Ph (K) the Set Of All Pluriharmonic Mappingsmentioning
confidence: 99%