| Paper title: |
On Certain Class of Harmonic Univalent Functions |
| Published in: | Issue 1, (Vol. 6) / 2012 |
| Publishing date: | 2011-04-11 |
| Pages: | 54-59 |
| Author(s): | AL KASBI Nasser, DARUS Maslina |
| Abstract. | A complex-valued functions that are univalent and sense preserving in the unit disk U can be written in the form f(z)= h(z)+ g(z), where U(z) and g(z) are analytic in. We will introduced the operator Dn which defined by convolution involving the polylogarithms functions. Using this operator, we introduce the class HP(α,γ, n) by generalized derivative operator of harmonic univalent functions. We give sufficient coefficient conditions for normalized harmonic functions in the class HP(α,γ, n). These conditions are also shown to be necessary when the coefficients are negative. This leads to distortion bounds and extreme points. |
| Keywords: | Univalent Functions, Harmonic Functions, Convex Combinations, Distortion Bounds |
| References: | 1.. H. Silverman, Harmonic univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51, (1998), 283-289. 2.. H. Silverman and E. M. Silvia, Subclasses of harmonic univalent functions, New Zeal. J. Math. 28, (1999), 275- 284. 3.. J. Clunie and T. Shell-Small, Harmonic univalent functions, Ann. Acad. Aci. Fenn. Ser. A I Math. 9, (1984), 3-25. 4.. J. M. Jahangiri, Harmonic functions starlike in the unit disk, J. Math. Anal. Appl. 235, (1999), 470-477. 5. K. Al-Shaqsi and M. Darus, On Certain Class of Harmonic Univalent Functions, Int. J. Contemp. Math. Sciences, Vol. 4, 2009, no. 24, 1193 – 1207 6.. St. Ruscheweyh, Neighborhood of univalent functions, Proc. Amer. Math. Soc. 81, (1981), 521-528. |
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