harmonic functions defined by generalised

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harmonic functions defined by generalised

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Jurnal Pengukuran Kualiti dan Analisis. HARMONIC FUNCTIONS DEFINED BY ... Ditunjukkan bahawa syarat pekali ini adalah perlu apabila fungsi berpekali ...

Journal of Quality Measurement and Analysis Jurnal Pengukuran Kualiti dan Analisis

JQMA 4(2) 2008, 101-111

HARMONIC FUNCTIONS DEFINED BY GENERALISED RUSCHEWEYH OPERATOR AND HADAMARD PRODUCTS (Fungsi Harmonik Tertakrif oleh Pengoperasi Terbitan Ruscheweyh Teritlak dan Hasil Darab Hadamard) KHALIFA AL-SHAQSI & MASLINA DARUS

ABSTRACT In this paper, we study the generalised Ruscheweyh derivatives operator D λn for harmonic function f . We introduce a class Mλn (A , B , α ) of all functions f , A and B for which ⎧ D n +1 ( f ∗ A )( z ) ⎫ Re ⎨ λ n ⎬ > α, for n ∈ N 0 , λ ≥ 0, 0 ≤ α < 1, ⎩ Dλ ( f ∗ B )( z ) ⎭

where ∗ denotes the Hadamard product (or convolution). We give sufficient coefficients conditions for normalised harmonic functions to be in Mλn (A , B , α ) . These conditions are also shown to be necessary when the coefficients are negative. Distortion theorems, extreme points, closure theorems and neighbourhood property are obtained. Keywords: Ruscheweyh derivatives; Hadamard product; Harmonic functions ABSTRAK Dalam makalah ini, pengoperasi terbitan Ruscheweyh teritlak D λn dikaji untuk fungsi harmonik f. Kelas Mλn (A , B , α ) diperkenalkan bagi semua fungsi f , A dan B yang memenuhi ⎧ D n +1 ( f ∗ A )( z ) ⎫ Re ⎨ λ n ⎬ > α, untuk n ∈ N 0 , λ ≥ 0, 0 ≤ α < 1, ⎩ Dλ ( f ∗ B )( z ) ⎭

yang ∗ melambangkan hasil darab Hadamard (atau konvolusi). Syarat cukup pekali bagi fungsi harmonik ternormal untuk berada dalam kelas Mλn (A , B , α ) diberikan. Ditunjukkan bahawa syarat pekali ini adalah perlu apabila fungsi berpekali negatif. Teorem erotan, titik ekstrim, teorem tutupan dan sifat kejiranan juga diperoleh. Kata kunci: terbitan Ruscheweyh; hasil darab Hadamard; fungsi harmonik

References Ahuja O. 1985. Integral operators of certain univalent functions. Internat. J. Math. Sci. 8: 653-662. Clunie J. & Sheil-Small T. 1984. Harmonic univalent functions. Ann. Acad. Sci. Fenn. Ser. A I Math. 9: 3-25. Darus M. & Al-Shaqsi K. 2006. On harmonic univalent functions defined by a generalized Ruscheweyh derivatives operator. Lobachevskii J. Math. 22: 19-26. Darus M. & Siregar S. 2005. Certain subclass of harmonic functions using Hadamard Product. Proc. Int. Advanced Tech. Congress. ITMA(Univ. Putra Malaysia) pp. 1-6. Jahangiri J.M. 1999. Harmonic functions starlike in the unit disk. J. Math. Anal. Appl. 235: 470-477. Murugusundaramoorthy G. 2003. A class of Ruscheweyh-type harmonic univalent functions with varing argument. South. J. Pure Appl. Math. 2: 90-95. Ruscheweyh S. 1981. Neighborhoods of univalent functions. Proc. Amer. Math. Soc. 81: 521-527. Sheil-Small T. 1990. Constants for planar harmonic mapping. J. London Math. Soc. 2(42): 237-248.

K. Al-Shaqsi & Maslina Darus

Silverman H. 1998. Harmonic univalent functions with negative coefficients. J. Math. Anal. Appl. 220: 283-289. Silverman H. & Silvia E. M. 1999. Subclasses of harmonic univalent functions. New Zea. J. Math. 28: 275-284.

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