Sains Malaysiana 40(4)(2011): 385–389

The Fekete-Szegö Theorem for a Certain Class of Analytic Functions

(Teorem Fekete-Szegö Bagi Suatu Kelas Fungsi Analisis)

Ma`Moun Harayzeh Al-Abbadi & Maslina Darus*

School of Mathematical Sciences, Faculty of Science and Technology

Universiti Kebangsaan Malaysia, Bangi 43600 Selangor D.E., Malaysia

Received: 9 December 2009 / Accepted: 18 July 2010

ABSTRACT

In this paper, we discuss a well known class studied by many authors including Ramesha et al. and Janteng, few to mention. Next, we extend the class to a wider class of functions f denoted by , which are normalized and univalent, in the open unit disk D={z:|z|<1} satisfying the condition:

where g ∈ S* (b),g(z) ≠ 0 is a normalized starlike function of order b, for 0 ≤ b < 1. For f ∈  we shall obtain sharp upper bounds for the Fekete-Szegö functional |a₃ – μ | when μ is real.

Keywords: Close-to-convex functions; convex functions; Fekete-Szegö theorem; starlike functions; univalent functions

ABSTRAK

Dalam kertas kerja ini, kelas yang terkenal yang dikaji oleh ramai penulis akan dibincangkan, termasuklah beberapa nama seperti Ramesha et al. dan Janteng. Seterusnya, kelas ini diperluaskan kepada kelas fungsi f dilambangkan oleh , yang ternormal dan univalen dalam cakera unit terbuka D={z:|z|<1} memenuhi syarat:

yang g ∈ S* (b),g(z) ≠ 0 adalah fungsi bak-bintang ternormal peringkatb, (0 ≤ b < 1). Untuk f ∈ uab batas atas terbaik bagi fungsian Fekete-Szegö |a₃ – μ | diperoleh apabila μ adalah nyata.

Kata kunci: Fungsi bak-bintang; fungsi cembung; fungsi hampir cembung; fungsi univalen; teorem Fekete-Szegö

REFERENCES

Akbarally, A. 2001. Beberapa Masalah Pekali dan Subordinasi untuk Fungsi Bak Bintang. MSc Thesis, Universiti Kebangsaan Malaysia, Malaysia.

Darus, M. & Thomas, D.K. 1996. On the Fekete-Szegö theorem for close-to-convex functions. Math. Japonica 44(3): 507-511.

Darus, M. & Thomas, D.K. 2000. The Fekete-Szegö problem for strongly close-to-convex functions. Scientiae Mathematicae3(2): 201-212.

Darus, M. 2002. The Fekete-Szegö problems for close-to-convex functions of the class Khs(α, β). Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis18(1): 13-18.

Darus, M. & Tuneski, N. 2003. On the Fekete-Szegö problem for generalized close-to-convex functions. Inter.Math. Jour. 4(6): 561-568.

Darus, M. & Hong, N.C. 2004. The Fekete-Szegö problem for strongly close-to-convex functions with Hadamard product. Far East J. Math. Sci. (FJMS) 15(1): 1-7.

Fekete, M. & Szegö, G. 1933. Eine Bermerkung über ungerade schlichte function. J. Lond. Math. Soc. 8: 85-89.

Frasin, B.A. & Darus, M. 2003. On the Fekete- Szegö using the Hadamard product. Internat. Math. J 3(12): 1297-1304.

Ibrahim, A. & Darus, M. 2001. The Fekete-Szegö theorem for close-to-convex functions: To determine the upper bounds. Jour. Inst. Math. Comp.Sci. (Math.Ser) 14(3): 207-216.

Janteng, A. 2006. Assortment of Problems for Certain Classes of Analytic Functions. PhD thesis, University of Malaya, K.L. Malaysia.

Janteng, A. 1999. Beberapa Masalah untuk Fungsi Bak Bintang. MSc thesis, Universiti Kebangsaan Malaysia, Malaysia.

Keogh, F.R. & Merkes, E.P. 1969. A coefficient inequality for certain classes of analytic functions. Proc. Amer. Math. Soc. 20: 8-12.

Koepf, W. 1987. On the Fekete-Szegö problem for close-to-convex functions. Proc. Amer. Math. Soc. 101: 89-95.

Obradovic, M. & Joshi, S.B. 1998. On certain classes of strongly starlike functions. Taiwanese J. Math 2: 297-302.

Pommerenke, C.H. 1975. Univalent Functions. Göttingen: Vandenhoeck and Ruprecht.

Ramesha, C. Kumar, S. & Padmanabhan, K.S. 1995. A sufficient condition for starlikeness. Chinese J. Math 23: 167-171.

Rahman, K.A. & Darus, M. 2001. The Fekete-Szegö problems for close-to-convex functions of the class Khs(α, β). Jour. Inst. Math. Comp. Sci.(Math. Ser) 14(3): 171-177.

*Corresponding authors; email: maslina@ukm.my