A new approach for the univalence of certain integral of harmonic mappings

Arbeláez, Hugo; Bravo, Víctor; Hernández, Rodrigo; Sierra, Willy; Venegas, Osvaldo

doi:10.1016/j.indag.2020.04.002

Cited by 5 publications

(10 citation statements)

References 26 publications

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“…By using this inequality, we can improve the last theorem. Proof The proof is completely analogous to that of the previous theorem; it is enough to replace the second condition in (8) with the inequality…”

Section: Zh(z) G(z)mentioning

confidence: 89%

“…Since the work by Clunie and Sheil-Small [13], many problems of geometric function theory have been extended from the setting of holomorphic functions to the wider class of harmonic mappings in the plane. In this direction, in [11] and subsequently in [8], the authors proposed an extension of the integral transforms (1) and (2) to the setting of sensepreserving harmonic mapping, see also Theorems 2 and 3 in [6]. The definitions given in [8] make use of the shear construction introduced by Clunie and Sheil-Small in [13] as follows: let f = h + g be a sense-preserving harmonic mapping in the unit disk D = {z ∈ C : |z| < 1} with the usual normalization g(0) = h(0) = 1h (0) = 0 and dilatation ω = g /h .…”

Section: Introductionmentioning

confidence: 92%

“…The authors proved in [8,Lemma 3] that if ϕ is a univalent analytic function in a linear invariant family F with order β, then…”

Section: Zh(z) G(z)mentioning

confidence: 99%

“…In this section, we propose an extension of the integral transforms (1) and (2) to nonvanishing logharmonic mappings in D. To this end, we will use the fact that if f is a nonvanishing logharmonic mapping in D, there is a branch of log f , which is harmonic in D with dilatation ω log f = ω f , and apply to it the theory that we developed in [8]. Note that ω log f = ω f implies that log f is a sense-preserving harmonic mapping and therefore J f is positive in D.…”

Section: Integral Transforms Of Non-vanishing Logharmonic Mappingsmentioning

confidence: 99%

“…Given α ∈ D, when ϕ = hg is zero only at z = 0, we define F α as the horizontal shear of with H(0) = G(0) = 0. In a similar way we extend the integral transform (2) to sensepreserving harmonic mappings in D. The reader can find interesting results, some of which are related to the question of knowing for which values of α the integral transforms lead functions belonging to the class S (or to some of its subclasses) to the class S in [8]. It is important to point out that the analog of the Bieberbach conjecture for sense-preserving univalent harmonic mappings f , proposed by Clunie and Sheil-Small [13], remains open even for the second Taylor coefficient a 2 of the analytic part h of f .…”

Section: Introductionmentioning

confidence: 99%

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Integral transforms for logharmonic mappings

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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.

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Section: Zh(z) G(z)mentioning

confidence: 89%