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Coefficient conditions for harmonic univalent mappings and hypergeometric mappings. (English) Zbl 1298.30001

Summary: In this paper, we obtain coefficient criteria for a normalized harmonic function defined in the unit disk to be close-to-convex and fully starlike, respectively. Using these coefficient conditions, we present different classes of harmonic close-to-convex (respectively, fully starlike) functions involving Gaussian hypergeometric functions. In addition, we present a convolution characterization for a class of univalent harmonic functions discussed recently by Mocanu, and later by Bshouty and Lyzzaik in 2010. Our approach provides examples of harmonic polynomials that are close-to-convex and starlike, respectively.

MSC:

30A05 Monogenic and polygenic functions of one complex variable
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
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References:

[1] Y. Abu Muhanna and S. Ponnusamy, Extreme points method and univalent harmonic mappings , · Zbl 1350.31003
[2] O.P. Ahuja, J.M. Jahangiri and H. Silverman, Convolutions for special classes of harmonic univalent functions , Appl. Math. Lett. 16 (2003), 905-909. · Zbl 1051.30015
[3] G.D. Anderson, M.K. Vamanamurthy and M. Vuorinen, Conformal invariants, inequalities, and quasiconformal maps , John Wiley & Sons, New York, 1997. · Zbl 0885.30012
[4] D. Bshouty and W. Hengartner, Univalent harmonic mappings in the plane. Handbook of complex analysis : Geometric function theory , Vol. 2, Kühnau, ed., Elsevier, Amsterdam, 2005. · Zbl 1074.30015
[5] D. Bshouty and A. Lyzzaik, Close-to-convexity criteria for planar harmonic mappings , Comp. Anal. Oper. Theor. 5 (2011), 767–774. · Zbl 1279.30011
[6] M. Chuaqui, P. Duren and B. Osgood, Curvature properties of planar harmonic mappings , Comp. Meth. Funct. Theor. 4 (2004), 127–142. · Zbl 1051.30025
[7] J.G. Clunie and T. Sheil-Small, Harmonic univalent functions , Ann. Acad. Sci. Fenn. 9 (1984), 3-25. · Zbl 0506.30007
[8] P. Duren, Harmonic mappings in the plane , Cambr. Tracts Math. 156 , Cambridge University Press, Cambridge, 2004. · Zbl 1055.31001
[9] J.M. Jahangiri, Harmonic functions starlike in the unit disk , J. Math. Anal. Appl. 235 (1999), 470-477. · Zbl 0940.30003
[10] J.M. Jahangiri, Ch. Morgan and T.J. Suffridge, Construction of close-to-convex harmonic polynomials , Comp. Var. Ellipt. Equat. 45 (2001), 319-326. · Zbl 1090.30501
[11] P.T. Mocanu, Sufficient conditions of univalency for complex functions in the class \(C^{1}\) , Anal. Num. Th. Approx. 10 (1981), 75-79. · Zbl 0481.30014
[12] P.T. Mocanu, Injectivity conditions in the compelex plane , Comp. Anal. Oper. Th. 5 (2011), 759-766. · Zbl 1279.30020
[13] S. Ponnusmay and J. Qiao, Univalent harmonic mappings with integer or half-integer coefficients, preprint , http://arxiv.org/abs/1207.3768.
[14] S. Ponnusamy and S. Rajasekaran, New sufficient conditions for starlike and univalent functions , Soochow J. Math. 21 (1995), 193-201. · Zbl 0843.30024
[15] S. Ponnusamy and A. Rasila, Planar harmonic and quasiregular mappings , in Topics in modern function theory , St. Ruscheweyh and S. Ponnusamy, eds., RMS-Lect. Notes Ser. 19 (2013), 267-333. · Zbl 1318.30039
[16] S. Ponnusamy and V. Singh, Univalence of certain integral transforms , Glasn. Matem. 31 (1996), 253-261. · Zbl 0870.30009
[17] S. Ponnusamy, H. Yamamoto and H. Yanagihara, Variability regions for certain families of harmonic univalent mappings , Comp. Var. Ellipt. Equat. 58 (2013), 23-34. · Zbl 1294.30045
[18] H. Silverman, Univalent functions with negative coefficients , J. Math. Anal. Appl. 220 (1998), 283-289. · Zbl 0908.30013
[19] T.J. Suffridge, Harmonic univalent polynomials , Compl. Var. Ellipt. Equat. 35 (1998), 93-107. · Zbl 0893.30005
[20] N.M. Temme, Special functions. An introduction to the classical functions of mathematical physics , Wiley, New York, 1996. · Zbl 0856.33001
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