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

Diserahkan: 9 Disember 2009 / Diterima: 18 Julai 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

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*Pengarang untuk surat-menyurat; email: maslina@ukm.my