Some results on a starlike log-harmonic mapping of order alpha

Aydoğan, Melike

doi:10.1016/j.cam.2013.07.008

Cited by 7 publications

(3 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The study of logharmonic functions was initiated mainly by the works of Abdulhadi, Bshouty, and Hengartner [3,5], and later continued in a series of papers in which the basic theory of logharmonic functions has been developed. Since then many papers have been published dealing with this subject, see for example [1,4,9,22,23].…”

Section: Some Preliminaries About Logharmonic Mappingsmentioning

confidence: 99%

Integral transforms for logharmonic mappings

et al. 2021

View full text Add to dashboard Cite

Bieberbach’s conjecture was very important in the development of geometric function theory, not only because of the result itself, but also due to the large amount of methods that have been developed in search of its proof. It is in this context that the integral transformations of the type $f_{\alpha }(z)=\int _{0}^{z}(f(\zeta )/\zeta )^{\alpha }\,d\zeta $ f α ( z ) = ∫ 0 z ( f ( ζ ) / ζ ) α d ζ or $F_{\alpha }(z)=\int _{0}^{z}(f'(\zeta ))^{\alpha }\,d\zeta $ F α ( z ) = ∫ 0 z ( f ′ ( ζ ) ) α d ζ appear. In this note we extend the classical problem of finding the values of $\alpha \in \mathbb{C}$ α ∈ C for which either $f_{\alpha }$ f α or $F_{\alpha }$ F α are univalent, whenever f belongs to some subclasses of univalent mappings in $\mathbb{D}$ D , to the case of logharmonic mappings by considering the extension of the shear construction introduced by Clunie and Sheil-Small in (Clunie and Sheil-Small in Ann. Acad. Sci. Fenn., Ser. A I 9:3–25, 1984) to this new scenario.

show abstract

Section: Some Preliminaries About Logharmonic Mappingsmentioning

confidence: 99%

Integral transforms for logharmonic mappings

et al. 2021

View full text Add to dashboard Cite

show abstract

“…where ω is an analytic function from D into itself, which is called second complex dilatation of f. The study of logharmonic functions was initiated mainly by the works of Abdulhadi, Bshouty, and Hengartner [2,5], and later continued in a series of papers in which is developed the basic theory of logharmonic functions. Since then many papers have been published dealing with this subject, see for example [3,4,6,17,18]. Because of the condition on ω, the Jacobian J f of f, given by…”

Section: Some Preliminaries About Logharmonic Mappingsmentioning

confidence: 99%

Integral transform for Logharmonic mappings

Arbeláez,

Bravo,

Hernández

et al. 2020

Preprint

View full text Add to dashboard Cite

Bieberbach's conjecture was very important in the development of Geometric Function Theory, not only because of the result itself, but also due to the large amount of methods that have been developed in search of its proof, it is in this context that the integral transformations of the typeIn this notes we extend the classical problem of finding the values of α ∈ C for which either f α or F α are univalent, whenever f belongs to some subclasses of univalent mappings in D, to the case of logharmonic mappings, by considering the extension of the shear construction introduced by Clunie and Sheil-Small in [11] to this new scenario.

show abstract

“…Note that f (0) = 0 if and only if m = 0, and that a univalent logharmonic mapping on U vanishes at the origin if and only if m = 1, that is, f has the form f (z) = z|z| 2β h(z)g(z), where Re(β) > −1/2 and 0 / ∈ (hg)(U ). This class has been studied extensively in recent years, for instance in [1,2,3,4,5,6,7,8,9,11,12,13,21,22,26] .…”

Section: Introductionmentioning

confidence: 99%

On geometrical properties of starlike logharmonic mappings

Abdulhadi¹,

Hajj²

2018

Journal of Classical Analysis

View full text Add to dashboard Cite

Abstract. In this paper, we find the radius of the disk Ω r such that every starlike logharmonic mapping f (z) of order α, is starlike in |z| r with respect to any point of Ω r . We also establish a relation between the set of starlike logharmonic mappings and the set of starlike logharmonic mappings of order α . Moreover, the radius of starlikeness and univalence for the set of close to starlike logharmonic mappings of order α is determined.Mathematics subject classification (2010): Primary 30C35, 30C45, Secondary 35Q30.

show abstract

Some results on a starlike log-harmonic mapping of order alpha

Cited by 7 publications

References 1 publication

Integral transforms for logharmonic mappings

Integral transforms for logharmonic mappings

Integral transform for Logharmonic mappings

On geometrical properties of starlike logharmonic mappings

Contact Info

Product

Resources

About