The Fekete-Szegö Theorem for a Certain Class of Analytic Functions (Teorem Fekete-Szegö Bagi Suatu Kelas Fungsi Analisis)

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1 Sains Malaysiana 40(4)(2011): 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* ABSTRACT In this paper, we discuss a well known class studied y many authors including Ramesha et al Janteng, few to mention Next, we extend the class to a wider class of functions f denoted y u a, which are normalized univalent, in the open unit disk D={z: z <1} satisfying the condition: where g S * (),g(z) 0 is a normalized starlike function of order, for 0 < 1 For f u a we shall otain sharp upper ounds for the Fekete-Szegö functional a 3 µa 22 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 diincangkan, termasuklah eerapa nama seperti Ramesha et al dan Janteng Seterusnya, kelas ini diperluaskan kepada kelas fungsi f dilamangkan oleh u a, yang ternormal dan univalen dalam cakera unit teruka D={z: z <1} memenuhi syarat: yang g S * (),g(z) 0 adalah fungsi ak-intang ternormal peringkat, (0 < 1) Untuk f u a atas atas teraik agi fungsian Fekete-Szegö a 3 µa 22 diperoleh apaila µ adalah nyata Kata kunci: Fungsi ak-intang; fungsi cemung; fungsi hampir cemung; fungsi univalen; teorem Fekete- Szegö INTRODUCTION AND DEFINITION Let S denotes the class of normalized analytic univalent functions f of the form: (1) a n is a complex numer z D = {z: z <1} A classical theorem of Fekete & Szegö (1933), states that for f S given y (1), (1987), Darus Thomas (1996, 2000), Irahim Darus (2001), Rahman Darus (2001), Darus (2002), Darus Tuneski (2003), Darus Hong (2004), Frasin Darus (2003) In particular, for f K e given y (1), Keogh Merkes (1969) showed that: for 0 μ 1 the inequality is sharp For the suclass of S, consisting of convex functions C, starlike functions S *, close-to-convex functions K Sharp upper ounds for the functional a 3 μa 22 have een studied y various authors, however we mention only few names here such as Keogh Merkes (1969), Koepf for each µ there is a function in K for which equality holds In this paper, we give an estimate for the same functional for the class u a defined as follows:

2 386 Definition 11: For 0 a<1 0 β<1, let the function f e analytic in D e given y (1) Then the function f u α β if only if there exist g S * (β) such that for z D (2) Here, S * (β) denotes the class of starlike functions of order β, that is g S * (β) if only if g is analytic in D, 2 3 = (p p 1 ) (1 β) (6) It also follows from (2) that α z 2 f" (z) + zf (z) = g(z) h(z) (7) where Re h(z) > 0, writing h(z) = 1 + c 1 z + c 2 z 2, where c 1, c 2,, are complex numers from (7), we have (3) Note that if the function g(z) in (2) is replaced y f (z), we get the definition of Ramesha et al (1995) However, the authors (Ramesha et al) did not define the inequality as in (3) The work of Ramesha et al (1995) has attracted many authors such as Oradovic Joshi (1998), Janteng (1999, 2006) Akarally (2001) We first state some preliminary lemmas, required for proving our result PRELIMINARY RESULTS Lemma 21: (Pommerenke 1975) Let h e analytic in D with Re h(z) > 0 e given y, h(z) = 1+c 1 z+c 2 z 2 +, for z D, then c n 2, where n 1, Lemma 22: (Darus & Thomas 1996) For 0 β<1, let g S *( β ) with g(z) = z + 2 z z 3 Then, for µ real And equating coefficients, gives 2(α + 1)a 2 = c (8) 3(2α + 1) a 3 = c c (9) The result now follows on using classical inequalities, p 1 2, p 2 2, c 1 2, c 2 2, applying the following inequalities, 2 2 (1 β) 3 (1 β) (3 2β), which follow from (5) (6) Now, we shall show our main result for the class u αβ MAIN RESULT Theorem 31: Let the function f e given y (1) elongs to the class u αβ Then, for 0 α < 1 0 β < 1, 3(2α + 1)(a + 1) 2 a 3 u a 22 3 µ 22 (1 β)max{1, 3 2β 4µ(1 β) } Next, we give our first result as follows: Lemma 23: Let the function f given y (1) elongs to the class u αβ Then (α + 1) a 2 2 β 3(2α + 1) a 3 (3 2β)(3 β) Proof Since g S * (β), it follows from (3) that zg (z) = [β + (1 β) p(z)] g(z) (4) For z β, with Re p(z) < 0 given y where p 1, p 2,, are complex numers Equating coefficients, we otain 2 (1 β), (5) Inequalities are sharp for all cases

3 387 Proof Write: From (8) (9), we have: A straightforward calculation, we find: Then we get: Now we have: Next, since x o 2, we get, the interval (12) (10) in the case From (10) gives: And hence, the result (12) completes the proof for the case: Now consider the first case for all We can see that the result is sharp y choosing: Having, c 2 = 2, p 1 = 2, p 2 = 2, 2 = 2(1 β) 3 = (3 2β) (1 β), sustituting in (10) to get the desired result Secondly, we consider the case: 2(α + 1) 2 3(2α + 1) μ 0, y applying the following inequalities, Write, 2 2(1 β), 3 (3 2 β) (1 β) Lemma 21 we get: Now we say that, y using the inequality we otain the following: (11) = φ(x), say, with x = c 1 Elementary calculation indicates that the function φ attains its maximum value at:

4 388 This completes the proof for this case Here, in order to find the upper ound for: (13) 2 A straightforward calculation shows that the function ψ(x) attains asolute maximum value in the interval [0, 2] at x o = 0 And after some simplifications, we conclude that ψ(x), 3 β, at x o = 0 And therefore, we get the result: (14) we use the previous result of (12), which is already proved for:, When sustitute in last result (12) gives the following: Hence completing the proof for the particular case After that, we have to make sure that the result is sharp for this case Upon choosing, c 1 = c 2 = p 2, 2 = 2(1 β) 3 = (3 2β) (1 β) in (10) we get the required result Next we consider the case for all: The result is sharp on choosing c 1 = 0, p 2 = 2, c 2 = 2, 2 = 0 3 = 1 β sustituting in (10), doing a simple calculation, we shall get the desired equality Finally, consider: Consider the case: Write: First, we take the case: We start from, (11) y sustituting: Now we say that, y using the inequality we otain the following: We get: Hence the proof is complete for this case Here, (15) = ψ(x), say, with x = c 1 Assume our maximum value, otained in x o [0, 2] ut through calculation it happens on so it is out of the interval, so it contradicts the assumption Then the maximum value may occur on the oundary x o = 0 or we used the last result (14) already proven for The result is sharp on choosing c 2 = 2i, c 2 = p 2 = -2, 2 = 2i(1 β) 3 (3 2β) (β 1) sustituting in (10)

5 389 ACKNOWLEDGEMENTs This work is supported y UKM-ST-06-FRGS , Malaysia The authors would like to thank referees for some comments that improved the paper REFERENCES Akarally, A 2001 Beerapa Masalah Pekali dan Suordinasi untuk Fungsi Bak Bintang MSc thesis, Universiti Keangsaan Malaysia, Malaysia Darus, M & Thomas, DK 1996 On the Fekete-Szegö theorem for close-to-convex functions Math Japonica 44(3): Darus, M & Thomas, DK 2000 The Fekete-Szegö prolem for strongly close-to-convex functions Scientiae Mathematicae 3(2): Darus, M 2002 The Fekete-Szegö prolems for close-to-convex functions of the class K hs (α, β) Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis 18(1): Darus, M & Tuneski, N 2003 On the Fekete-Szegö prolem for generalized close-to-convex functions InterMath Jour 4(6): Darus, M & Hong, NC 2004 The Fekete-Szegö prolem 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 üer ungerade schlichte function J Lond Math Soc 8: Frasin, BA & Darus, M 2003 On the Fekete- Szegö using the Hadamard product Internat Math J 3(12): Irahim, A & Darus, M 2001 The Fekete-Szegö theorem for close-to-convex functions: To determine the upper ounds Jour Inst Math CompSci (MathSer) 14(3): Janteng, A 2006 Assortment of Prolems for Certain Classes of Analytic Functions PhD thesis, University of Malaya, KL Malaysia Janteng, A 1999 Beerapa Masalah untuk Fungsi Bak Bintang MSc thesis, Universiti Keangsaan Malaysia, Malaysia Keogh, FR & Merkes, EP 1969 A coefficient inequality for certain classes of analytic functions Proc Amer Math Soc 20: 8-12 Koepf, W 1987 On the Fekete-Szegö prolem for close-toconvex functions Proc Amer Math Soc 101: Oradovic, M & Joshi, SB 1998 On certain classes of strongly starlike functions Taiwanese J Math 2: Pommerenke, CH 1975 Univalent Functions Göttingen: Venhoeck Ruprecht Ramesha, C Kumar, S & Padmanahan, KS 1995 A sufficient condition for starlikeness Chinese J Math 23: Rahman, KA & Darus, M 2001 The Fekete-Szegö prolems for close-to-convex functions of the class K hs (α, β) Jour Inst Math Comp Sci(Math Ser) 14(3): School of Mathematical Sciences Faculty of Science Technology Universiti Keangsaan Malaysia Bangi, Selangor DE Malaysia *Corresponding authors; maslina@ukmmy Received: 9 Decemer 2009 Accepted: 18 July 2010