SOME RESULTS ON A SUBCLASS OF HARMONIC MAPPINGS OF ORDER ALPHA

Year 2014, Volume: 4 Issue: 1, 104 - 109, 01.06.2014

D. Varol Melike Aydogan S. Owa

Abstract

Let SH be the class of harmonic mappings defined by SH = { f = h z + g z | h z = z + ∑∞ n=2 anz n , g z = b1z + ∑∞ n=2 bnz n , b1 < 1 } where h z and g z are analytic. Additionally f z ∈ SH α ⇔     zh′ z − zg′ z h z + g z − 1 − b1 1 + b1 < 1 − b1 1 + b1  − α, z ∈ U, 0 6 α < 1 − b1 1 + b1 In the present work, by considering the analyticity of the functions defined by R. M. Robinson [7], we discuss the applications to the harmonic mappings.

Keywords

Harmonic Mappings, Subordination principle, Distortion theorem

References

• Clunie and T. Sheil-Small, Harmonic Univalent functions, Annales Academiae Scientiarum Fennicae, Series A, Vol. 9, pp. 3-35, (1984).
• Duren, P. , Harmonic Mappings in the Plane, Vol. 156 of Cambridge Tracts in Mathematics, Cam- bridge University Press, Cambridge UK, (2004).
• Goodman A. W. , Univalent Functions, Volume I and Volume II, Mariner publishing Company INC, Tampa Florida, (1983).
• R.J. Libera, Some Radius of Convexity Problem, Duke Math. J.31(1964) 143-157.
• W. Janowski, Some extremal problems for certain families of analytic functions I, Annales Polinici Mathematici 27, (1973), 298-326.
• M. A. Nasr and M. K. Aouf, Starlike functions of complex order, Jour. of Natural science and Math- ematics vol. 25, No.1, (1985), 1-12.
• Robinson, R. M . , Univalent majorants, Trans. Amer. Math. Soc. 61 (1947), 1-35.