Precise Coefficient Estimates for Close-to-Convex Harmonic Univalent Mappings

Wang, Xiao-Tian; Liang, Xian-Ting; Zhang, Yulin

doi:10.1006/jmaa.2001.7626

Cited by 44 publications

(19 citation statements)

References 2 publications

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“…In 1984, Clunie and Sheil-Small [6] investigated the class S H and its geometric subclasses. Since then, the class S H and its subclasses have been extensively studied (see [4], [5], [6], [14], [32]). A domain Ω is called starlike with respect to a point z 0 ∈ Ω if the line segment joining z 0 to any point in Ω lies in Ω.…”

Section: Introductionmentioning

confidence: 99%

On a subclass of harmonic close-to-convex mappings

Ghosh

Allu

2017

Monatsh Math

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Let H denote the class of harmonic functions f in D := {z ∈ C : |z| < 1} normalized by f (0) = 0 = f z (0) − 1. For α ≥ 0, we consider the following classIn this paper, we first prove the coefficient conjecture of Clunie and Sheil-Small for functions in the class W 0 H (α). We also prove growth theorem, convolution, convex combination properties for functions in the class W 0 H (α). Finally, we determine the value of r so that the partial sums of functions in the class W 0 H (α) are closeto-convex in |z| < r.1991 Mathematics Subject Classification. Primary 30C45, 30C80.

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Section: Introductionmentioning

confidence: 99%

On a subclass of harmonic close-to-convex mappings

Ghosh

Allu

2017

Monatsh Math

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“…(4) Although, the coefficient's conjecture remains an open problem for the full class S 0 H , the same has been verified for certain subclasses: namely, the class T H (see [2,Section 6.6]) of harmonic univalent typically real functions, the class of harmonic convex functions in one direction, harmonic starlike functions in S 0 H (see [2,Section 6.7]), and the class of harmonic close-to-convex functions (see [4]). …”

Section: Introduction and Main Resultsmentioning

confidence: 97%

“…Proof of Lemma 1.2 Let h and g have the form (1) satisfying the coefficient conditions (4). First, we observe that b 1 = g (0) = 0.…”

Section: Proofs Of Main Resultsmentioning

confidence: 98%

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Radius of close-to-convexity and fully starlikeness of harmonic mappings

Kalaj

Ponnusamy

Вуоринен

2013

Complex Variables and Elliptic Equations

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Let H denote the class of all normalized complex-valued harmonic functions f = h + g in the unit disk D, and let S 0 H denote the class of univalent and sense-preserving functions f in H such that f z (0) = 0. If K = H + G denotes the harmonic Koebe function whose dilation isHere, a n , b n , A n , and B n denote the Maclaurin coefficients of h, g, H, and G. We show that the radius of univalence of the family F is 0.112903 . . .. We also show that this number is also the radius of the fully starlikeness of F . Analogous results are proved for a family which contains the class of harmonic convex functions in H. We use the new coefficient estimate for bounded harmonic mappings and Lemma 1.6 to improve Bloch-Landau constant for bounded harmonic mappings.

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“…for all j ≥ 1. Although, this coefficient conjecture remains an open problem for the full class S 0 H , this statement has been verified for certain subclasses, namely, the class T H (see [12,Section 6.6]) of harmonic univalent typically real mappings, the class of harmonic convex mappings in one direction, harmonic starlike mappings in S 0 H (see [12,Section 6.7]), and the class of harmonic close-to-convex mappings (see [23]). Equality occurs in (1.2) for the harmonic Koebe mapping…”

Section: Introductionmentioning

confidence: 94%

Starlikeness and convexity of polyharmonic mappings

Chen¹,

Rasila²,

Wang³

2014

Bull. Belg. Math. Soc. Simon Stevin

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In this paper, we first find an estimate for the range of polyharmonic mappings in the class HC 0 p . Then, we obtain two characterizations in terms of the convolution for polyharmonic mappings to be starlike of order α, and convex of order β, respectively. Finally, we study the radii of starlikeness and convexity for polyharmonic mappings, under certain coefficient conditions. 2010 Mathematics Subject Classification. Primary 30C65, 30C45; Secondary 30C20.

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Precise Coefficient Estimates for Close-to-Convex Harmonic Univalent Mappings

Cited by 44 publications

References 2 publications

On a subclass of harmonic close-to-convex mappings

On a subclass of harmonic close-to-convex mappings

Radius of close-to-convexity and fully starlikeness of harmonic mappings

Starlikeness and convexity of polyharmonic mappings

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