On the univalence of certain integral for harmonic mappings

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

“…In the last years we have seen several papers about geometric function theory but in the context of harmonic mappings, since the seminal work of Clunie and Sheil-Small [3] and the excellent book of P. Duren [5]. In this sense, the present paper is part of the same effort and continues the work presented in [2].…”

“…Section 3 is devoted to extend the integral transformation of the first type given by equation 1 to harmonic mappings and to find the real values of α for which the corresponding function F α is,either univalent function or close-to-convex function, considering that ϕ belongs to different classes of univalent functions. Finally, in Section (4) we give an alternative extension of the integral transformation of the second type defined in (2). This extension is different from the one given in [2].…”

“…Finally, in Section (4) we give an alternative extension of the integral transformation of the second type defined in (2). This extension is different from the one given in [2].…”

On quasiconformal extensions of harmonic mappings associated with pre-Schwarzian derivative

“…The following result was derived by Bravo et al [5], which generalized Theorem H. It also can be regarded as a generalization of harmonic analogue of Theorem A.…”

Integral transforms for logharmonic mappings

“…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 .…”