On rotationally starlike logharmonic mappings

Abdulhadi, Zayid; Ali, Rosihan M.

doi:10.1002/mana.201400056

Cited by 7 publications

(2 citation statements)

References 19 publications

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“…The class of functions of this form has been widely studied. See, for example [3,8,14]. For simplicity, we set β = 0 and consider the class S Lh of univalent log-harmonic mappings f of the form f (z) = zh(z)g(z),…”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

Some properties of univalent log-harmonic mappings

Liu¹,

Ponnusamy²

2018

Filomat

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We consider the class univalent log-harmonic mappings on the unit disk. Firstly, we obtain necessary and sufficient conditions for a complex-valued continuous function to be starlike or convex in the unit disk. Then we present a general idea, for example, to construct log-harmonic Koebe mapping, log-harmonic right half-plane mapping and logharmonic two-slits mapping and then we show precise ranges of these mappings. Moreover, coefficient estimates for univalent log-harmonic starlike mappings are obtained. Growth and distortion theorems for certain special subclass of log-harmonic mappings are studied. Finally, we propose two conjectures, namely, log-harmonic coefficient and log-harmonic covering conjectures.2010 Mathematics Subject Classification. Primary: 30C35, 30C45; Secondary: 35Q30.

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Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

Some properties of univalent log-harmonic mappings

Liu¹,

Ponnusamy²

2018

Filomat

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“…where 0 / ∈ (hg)(U). This class has been studied extensively over recent years in [1][2][3][4][5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

Characterizing Rotationally Typically Real Logharmonic Mappings

Alarifi¹,

Abdulhadi²,

Ali³

2017

Preprint

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This paper treats the class of normalized logharmonic mappings f (z) = zh(z)g(z) in the unit disk satisfying ϕ(z) = zh(z)g(z) is analytically typically real. Every such mapping f is shown to be a product of two particular logharmonic mappings, each of which admits an integral representation. Also obtained is the radius of starlikeness and an upper estimate for arclength. Additionally, it is shown that f maps the unit disk into a domain symmetric with respect to the real axis when it is univalent and its second dilatation has real coefficients.

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