Abstract
In this paper, we introduce a new subclass of harmonic functions defined by shear construction. Among other properties of this subclass, we study the convolution of its elements of it with some special subclasses of harmonic functions. Also, we provide coefficient conditions leading to harmonic mappings which are starlike of order \(\alpha \). Some interesting applications of the general results are also presented.
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Clunie, J., Sheil-Small, T.: Harmonic univalent functions. Ann. Acad. Sci. Fenn. Ser. A I Math. 9, 3–25 (1984)
Droff, M.: Convolutions of plannar harmonic convex mappings. Complex Var. Theory Appl. 45, 263–271 (2001)
Goodman, A.W.: Univalent functions and nonanalytic curves. Proc. Am. Math. Soc. 8, 598–601 (1957)
Jahangiri, J.: Coefficient bounds and univalence criteria for harmonic functions with negative coefficient. Ann. Univ. Mariae Curie-Sklodowska Sect. A. 2, 57–66 (1998)
Jahangiri, J.: Harmonic functions starlike in the unit disk. J. Math. Anal. Appl. 235, 470–477 (1999)
Lewy, H.: On the non-vanishing of the Jacobian in certain one-to-one mappings. Bull. Am. Math. Soc. 42, 689–692 (1936)
Libera, R.J.: Some classes of regular univalent functions. Proc. Am. Math. Soc. 16, 755–758 (1965)
Muir, S.: Weak subordination for convex univalent harmonic function. J. Math. Anal. Appl. 348, 862–871 (2008)
Muir, S.: Harmonic mappings convex in one or every direction. Comput. Methods Funct. Theory. 12(1), 221–239 (2012)
Pokhrel, C.M.: Convexity preservation for analytic, harmonic and plane curves. Ph.D. thesis. Tribhuvan University, Kathmandu (2004)
Pommenrenke, C.: On starlike and close-to-convex functions. Proc. London Math. Soc. 13, 290–304 (1963)
Ruscheweyh, S., Sheil-Small, T.: Hadamard products of Schlicht functions and the Polya Schoenberg conjecture. Comment. Math. Helv. 48, 119–135 (1973)
Ruscheweyh, S., Salinas, L.: On the preservation of direction-convexity and the Goodman-Saff conjecture. Ann. Acad. Sci. Fenn. 14, 63–73 (1989)
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The authors would like to thank the referees for their careful reading of the paper and for their helpful comments to improve it.
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Aghalary, R., Rad, M.J. Properties of Harmonic Functions Defined by Shear Construction. Acta Math Vietnam 43, 471–483 (2018). https://doi.org/10.1007/s40306-017-0235-y
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DOI: https://doi.org/10.1007/s40306-017-0235-y