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Combinations of Right Half-Plane Mappings and Vertical Strip Mappings Convex in the Vertical Direction
Department of Mathematics, Hanshan Normal University, Chaozhou, China
Department of Mathematics, Hanshan Normal University, Chaozhou, China
Department of Architecture, Foshan University, Foshan, China
Department of English, Hanshan Normal University, Chaozhou, China
- 1 Department of Mathematics, Hanshan Normal University, Chaozhou, China
- 2 Department of Mathematics, Hanshan Normal University, Chaozhou, China
- 3 Department of Architecture, Foshan University, Foshan, China
- 4 Department of English, Hanshan Normal University, Chaozhou, China
Open Journal of Statistics·Volume 13 (2023)·Pages 941–954·Published 13 November 2023·DOI10.4236/ojs.2023.136047
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Abstract
In this paper, we explore the linear combinations of right half-plane mappings and vertical strip mappings. We demonstrate that the combinations of these h armonic mappings are convex in the vertical direction provide d 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 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.
KeywordsHarmonic MappingLinear CombinationCohn’s RulesDirectional Convexi-ty
- Campbell, D.M. (1975) A Survey of Properties of the Convex Combination of Univalent Functions. The Rocky Mountain Journal of Mathematics, 5, 475-492. https://doi.org/10.1216/RMJ-1975-5-4-475
- Lewy, H. (1936) On the Non-Vanishing of the Jacobian in Certain One-to-One Mappings. Bulletin of the American Mathematical Society, 10, 689-693. https://doi.org/10.1090/S0002-9904-1936-06397-4
- Clunie, J. and Sheil-Small, T. (1984) Harmonic Univalent Functions. Annales Academiae Scientiarum Fennicae, 9, 3-25. https://doi.org/10.5186/aasfm.1984.0905
- Rahman, Q.I. and Schmeisser, G. (2002) Analytic Theory of Polynomials. Oxford University Press, Oxford, UK. https://doi.org/10.1093/oso/9780198534938.001.0001
- Hengartner, W. and Schober, G. (1970) On Schlicht Mappings to Domains Convex in One Direction. Commentarii Mathematic Helvetici, 45, 303-314. https://doi.org/10.1007/BF02567334
- Wang, Z.G., Liu, Z.H. and Li, Y.C. (2013) On the Linear Combinations of Harmonic Univalent Mappings. Journal of Mathematical Analysis and Applications, 400, 452-459. https://doi.org/10.1016/j.jmaa.2012.09.011
- Kumar, R., Gupta, S. and Singh, S. (2016) Linear Combinations of Univalent Harmonic Mappings Convex in the Direction of the Imaginary Axis. Bulletin of the Malaysian Mathematical Sciences Society, 39, 751-763. https://doi.org/10.1007/s40840-015-0190-5
- Long, B.Y. and Dorff, M. (2018) Linear Combinations of a Class of Harmonic Univalent Mappings. Filomat, 32, 3111-3121. https://doi.org/10.2298/FIL1809111L
- Zireh, A. and Shabani, M.M. (2019) On the Linear Combinations of Slanted Half-Plane Harmonic Mappings. Sahand Communications in Mathematical Analysis, 14, 89-96.
- Beig, S., Sim, Y.J. and Cho, N.E. (2020) On Convex Combinations of Harmonic Mappings. Journal of Inequalities and Applications, 2020, Article No. 84. https://doi.org/10.1186/s13660-020-02350-8
- Liu, Z.H., Cai, Y.L., Wang, Z.W., Shi, L. and Li, Y.C. (2021) Linear Combinations of Univalent Mappings with Complex Coefficients. Journal of Mathematics, 2021, Article ID: 4727617. https://doi.org/10.1155/2021/4727617
- Khurana, D., Kumar, R., Gupta, S. and Singh, S. (2022) Linear Combinations of Univalent Harmonic Mappings with Complex Coefficients. Matematicki Vesnik, 74, 189-196.