2020
DOI: 10.1186/s13660-020-02350-8
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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 uni… Show more

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Cited by 3 publications
(6 citation statements)
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References 13 publications
(15 reference statements)
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“…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.…”
mentioning
confidence: 86%
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“…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.…”
mentioning
confidence: 86%
“…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.…”
Section: Introductionmentioning
confidence: 99%
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“…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…”
Section: Motivation and Preliminariesmentioning
confidence: 99%
“…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 ( )…”
Section: Introductionmentioning
confidence: 99%