On convex combinations of harmonic mappings

Abstract

Let ψ μ,ν (z) = (1-2 cos νe iμ z + e 2iμ z 2)-1 , μ, ν ∈ [0, 2π) and p be an analytic mapping with Re p > 0 on the open unit disk. We consider the sense-preserving planar harmonic mappings f = h + g, which are shears of the mapping z 0 ψ μ,ν (ξ)p(ξ) dξ in the direction μ. These mappings include the harmonic right half-plan mappings, vertical strip mappings, and their rotations. For various choices of dilatations g /h of f , sufficient conditions are found for the convex combinations of these mappings to be univalent and convex in the direction μ.

Linear Combinations of Univalent Mappings with Complex Coefficients

“…Remark 2. When α � β + kπ(k ∈ Z) in eorem 5, since α � arg(λ) and β � arg(1 − λ), we obtain that λ is a real number in this case, and then it is reduced to [12], eorem 2.3. Now, we remove some restrictions and get the following two conclusions.…”

“…In this paper, motivated by the work carried out in [11][12][13], we study the linear combination f � λf 1 + (1 − λ)f 2 in the cases when λ takes some complex number.…”

“…In eorem 4, if we set μ � 0 and p 1 (z) � p 2 (z) � 1, it is reduced to eorem 2. Also, eorem 4 provides an extension of [12], eorem 2.1, for the case of a real number λ. Theorem 5. For j � 1, 2, let f j � h j + g j ∈ S H and satisfy…”

On Convolution and Convex Combination of Harmonic Mappings

“…It is known that even if f and g are convex analytic functions, the convex combination of f and g may not be a univalent function (see [18], and for more recent studies of linear combinations of harmonic mappings, see [17,[19][20][21][22][23]). Moreover, Kumar et al [20] studied the convexity of linear combination of harmonic mappings, which are shears of the analytic mapping [23] studied and found necessary conditions for the convex combination of the right half-plane mappings, the vertical strip mapping, their rotations, and some other harmonic mappings to be univalent and convex in a particular direction. In Section 3, the convex combination of mappings of the family of sense-preserving and locally univalent harmonic mappings f α � h α + g α , by shearing the function h α + g α � K α where…”

Combinations of Right Half-Plane Mappings and Vertical Strip Mappings Convex in the Vertical Direction

“…proved the sufficient conditions for the linear combinations of two slanted half-plane harmonic mappings to be univalent and convex in an arbitrary direction of γ − . In recent years, Beig, S. [10] et al have demonstrated that the linear combination of two different kinds of harmonic mappings is univalent and convex in a special direction γ , they have further generalized this result to more common cases by setting certain conditions. In this paper, inspired by the research conducted in [6] [7] [8] [9] [10], we investigate the linear combinations ( )…”