ON A CRITERION FOR MULTIVALENT HARMONIC FUNCTIONS

Year 2014, Volume: 4 Issue: 1, 50 - 55, 01.06.2014

T. Hayami

Abstract

For normalized harmonic functions f z = h z + g z in the open unit disk, a criterion on the analytic part h z for f z to be p-valent and sense-preserving is discussed. Furthermore, several illustrative examples and images of f z satisfying the obtained condition are enumerated.

Keywords

Harmonic function, Multivalent function, Univalent function

References

• Bshouty, D. and Lyzzaik, A., (2011), Close-to-convexity criteria for planar harmonic mappings, Com- plex Anal. Oper. Theory, 5, 767-774.
• Clunie, J. and Sheil-Small, T., (1984), Harmonic univalent functions, Ann. Acad. Sci. Fenn. Ser. A I Math., 9, 3-25.
• Duren, P. L., (2004), Harmonic Mappings in the Plane, Cambridge University Press, Cambridge.
• Goodman, A. W., (1983), Univalent Functions, Vol. I and Vol. II, Mariner Publishing Company, Tampa, Florida.
• Hayami, T. and Owa, S., (2011), Hypocycloid of n + 1 cusps harmonic function, Bull. Math. Anal. Appl., 3, 239-246.
• Hayami, T., (2012), Coefficient conditions for harmonic close-to-convex functions, Abstr. Appl. Anal., , Article ID 413965, 12 pp.
• Hayami, T., (2013), A sufficient condition for p-valently harmonic functions, Complex Var. Elliptic
• Equ., Published online. DOI : 10.1080/17476933.2013.826656.
• Lewy, H., (1936), On the non-vanishing of the Jacobian in certain one-to-one mappings, Bull. Amer. Math. Soc., 42, 689-692.
• Mocanu, P. T., (1981), Sufficient conditions of univalency for complex functions in the class C1, Rev. d’Anal. Nume´er. et de Th´eorie Approx., 10, 75-81.
• Mocanu, P. T., (2009), Three-cornered hat harmonic functions, Complex Var. Elliptic Equ., 54, 1079
• Mocanu, P. T., (2011), Injectively conditions in the complex plane, Complex Anal. Oper. Theory, 5, 766.