Linear Combinations of Univalent Mappings with Complex Coefficients

Liu, Zhihong; Cai, Yulin; Wang, Ziwei; Shi, Lei; Li, Yingchun

doi:10.1155/2021/9188781

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2023

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“…Let 3 ω be the dilatation of 3 f which satisfies the Equation (10). By setting 1 z ω = − , 2 z ω = , and substituting them into Equation (11), we obtain…”

Section: Resultsmentioning

confidence: 99%

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

Cai,

She,

Zeng

et al. 2023

OJS

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In this paper, we explore the linear combinations of right half-plane mappings and vertical strip mappings. We demonstrate that the combinations of these harmonic mappings are convex in the vertical direction provided they are locally univalent and sense-preserving. Furthermore, we extend this analysis to a more general case by setting specific conditions. Additionally, we take some common parameters such as , ,e i z z z θ − as the dilatation of these harmonic mappings, and prove the sufficient conditions that their combinations are locally univalent and convex in the vertical direction. Several examples are constructed by the Mathematica software to demonstrate our main results.

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“…Let 3 ω be the dilatation of 3 f which satisfies the Equation (10). By setting 1 z ω = − , 2 z ω = , and substituting them into Equation (11), we obtain…”

Section: Resultsmentioning

confidence: 99%