Pure Mathematical Sciences, Vol. 1, 01, no. 4, 187-196 The Fekete Szegö Theorem for a Subclass of Quasi-Convex Functions Goh Jiun Shyan School of Science and Technology Universiti Malaysia Sabah Jalan UMS, 88400 Kota Kinabalu Sabah, Malaysia Aini Janteng School of Science and Technology Universiti Malaysia Sabah Jalan UMS, 88400 Kota Kinabalu Sabah, Malaysia Abstract For 0 α<1, 0 β<1, 0 γ<1, let Kαβ, γ denote the class of functions f which are normalised and univalent in the open unit disk D = {z : z <1} satisfying the condition { αz f z Re g + zf z } z g >β, z where g S γ. For f Kαβ, γ, sharp bounds are obtained for the Fekete-Szegö functional a 3 µa when µ is real. Mathematics Subject Classification: Primary 30C45 Keywords: univalent, quasi-convex functions, Fekete-Szegö functional 1 Introduction Let S denote the class of normalised analytic univalent functions f of the form fz = z + a n z n, a n is complex number, 1 n=
188 Goh Jiun Shyan and Aini Janteng where z D = {z : z <1}. A classical theorem of Fekete and Szegö[9] states that for f S and given by 1,, a3 µa 1 + exp µ 1 µ for 0 µ 1 and the inequality is sharp. Many studies of functional a 3 µa have been done for the subclass of S, which are consisting of convex functions C, starlike functions S and close-to-convex functions K, namely Keogh and Merkes[4], Keopf[10], Darus and Thomas[6, 7, 8] and Frasin and Darus[]. For f K and be given by 1 and µ is real, Keogh and Merkes[4] found that 3 4µ, if µ 1, 3 1 a3 µa + 4, if 1 µ, 3 9µ 3 3 1, if µ 1, 3 4µ 3, if µ 1 and 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 our class K αβ, γ defined as follows: Definition 1.1 Let f be given by 1, 0 α<1, 0 β<1, and 0 γ<1. Then f Kαβ, γ if and only if there exists g Cγ such that for z D { αz f z Re + zf } z >β. g z g z Here Cγ denote the class of convex functions of order γ; that is g Cγ if and only if g is analytic in D and { } zg z Re >γ, z D. g z Definition above is also equivalent to the following: f Kαβ, γ if and only if there exists hz = zg z S γ such that { αzz f z Re + zzf } z >β hz hz We note that K α0, 0 = Q α, class of functions fz introduced by Janteng et al.[1] and K αβ, 0 = K αβ, class of functions fz introduced by Goh and Janteng[5].
Fekete Szegö theorem 189 Preliminary Results There are some preliminary lemmas required for proving our results. Lemma.1 [3] Let h be analytic in D with Re{hz}>0 and be given by hz = 1 + c 1 z + c z +... for z D, then c c 1 c 1. Lemma. [6] For 0 β<1, let g S β with gz = z + b z + b 3 z 3 +.... Then, for µ real, b 3 µb 1 βmax{1, 3 β 4µ1 β }. Lemma.3 Let f K αβ, γ, 0 α<1, 0 β<1, 0 γ<1 and be given by 1. Then 4α + 1 a γ β 3 and 9α + 1 a 3 1 γ7 γ 4β + 1 β 4 Proof. Since h S γ, it follows that zh z = [γ + 1 γpz] hz 5 for z D, with Re{pz}>0 given by pz = 1 + p 1 z + p z + p 3 z 3 +. Equating coefficients in 5, we obtain b = p 1 1 γ 6 and It also follows from that b 3 = 1 γ b p 1 + p. 7 αzz f z + zzf z = β + 1 βkz hz 8 where Re{kz}>0. Writing kz = 1 + c 1 z + c z + c 3 z 3 +. and equating coefficients in 8 gives 4α + 1a = b + 1 βc 1 9 and 9α + 1a 3 = b 3 + 1 βb c 1 + 1 βc. 10
190 Goh Jiun Shyan and Aini Janteng The result now follows by using classical inequalities, p 1, p, c 1, c, and applying the following inequalities, b 1 γ 11 and b 3 1 γ3 γ 1 which follow from 6 and 7. 3 Main Result The result on Fekete-Szegö functional for the class K αβ, γ, 0 α<1, 0 β<1 and 0 γ<1 is given by the following theorem. Theorem 3.1 Let f be given by 1 and belongs to the class K αβ, γ. Then, for 0 α<1, 0 β<1 and 0 γ<1. α + 1 1 γ7 γ 4β + 1 β 9α+1α+1 a 3 µa 9α+1 γ β µ, 4 if µ 81 γα+1 9 γ βα+1 α + 1 1 γ3 γ + 1 β 9α+11 γ µ + 1 γ [8α+1 9α+1µ], 4α+1 36α+1 α+1µ 81 γα+1 if µ 8α+1 9 γ βα+1 9α+1 α + 1 1 γ + 1 β, if 8α+1 µ 4α+1 1 γ8 γ 4β+41 β 9α+1 9α+1 γ β α + 1 1 γ7 γ 4β +1 β + 9α+1 γ β µ, 4 if µ 4α+1 1 γ8 γ 4β+41 β 9α+1 γ Inequalities are sharp for all cases.
Fekete Szegö theorem 191 Proof. From 6, 7, 9 and 10, it is easily established that { } 9α + 1a 3 µa 9α + 1µ = b 3 16α + 1 b { { 8α + 1 9α + 1µ + c 1 β + 1 } 16α + 1 } { } c 11 β 9α + 1µ + 1 b 8α + 1 c 1 1 β 13 81 γα+1 First, consider the case µ 8α+1. 9 γ βα+1 9α+1 Equation13gives 9α + 1 a 3 µa b 9α + 1µ 3 16α + 1 b + c 1 β 1 c 11 β + 8α + 1 9α + 1µ 16α + 1 c 1 1 β + b c 1 1 β 8α + 1 9α + 1µ 14 8α + 1 9α + 11 γµ 1 γ 3 γ 4α + 1 + 1 β 1 c 1 1 β + 1 16α + 1 8α + 1 9α + 1µ c 1 1 β + 1 γ1 β c 1 8α + 1 9α + 1µ 4α + 1 = 1 γ3 γ 9α + 11 γ µ + 1 β 4α + 1 9α + 1µ 16α + 1 c 1 1 β + 1 γ[8α + 1 9α + 1µ] 4α + 1 c 1 1 β 15 = øx, say, with x = c 1, where we have used Lemma.1, Lemma. and the inequality p 1. Elementary calculation indicates that the function ø attains its maximum value when x = x 0 = 1 γ[8α + 1 9α + 1µ], 91 βα + 1µ
19 Goh Jiun Shyan and Aini Janteng and thus establishing 9α + 1α + 1 a 3 µa α + 1 øx 0 = α + 1 1 γ3 γ + 1 β 9α + 11 γ µ 4α + 1 + 1 γ [8α + 1 9α + 1µ] 36α + 1 α + 1µ 16 Next, since x 0, thus we have µ the proof for the case 81 γα+1 9 γ βα+1 81 γα+1 9 γ βα+1 µ 8α+1 9α+1. and hence completing Letting c 1 = 1 γ[8α+1 9α+1µ] 91 βα+1µ, c =, p 1 = p =, b = 1 γ, b 3 = 1 γ3 γ in 13 shows that the result is sharp. Next, consider the case µ Write a 3 µa = a 3 81 γα+1. 9 γ βα+1 { } 81 γα + 1 81 γα + 1 9 γ βα + 1 a + 9 γ βα + 1 µ a By using the inequality a γ β α+1, 9α + 1α + 1 a 3 µa 9α + 1α + 1 81 γα + 1 a 3 9 γ βα + 1 a + 9α + 1α + 1 { } 81 γα + 1 9 γ βα + 1 µ a =α + 1 1 γ7 γ 4β + 1 β 9α + 1 γ β µ 4 81 γα+1 Here, we use the result already proven for. The result is sharp 9 γ βα+1 by choosing c 1 = c = p 1 = p =, b = 1 γ, b 3 = 1 γ3 γ in 13. Next, assume that 8α+1 9α+1 µ 4α+1 1 γ8 γ 4β+41 β 9α+1 γ β.
Fekete Szegö theorem 193 First, we deal with the case µ = 4α+1 1 γ8 γ 4β+41 β and from 15 9α+1 γ β we have 9α + 1 a 3 4α + 1 1 γ8 γ 4β + 41 β a 9α + 1 γ β 1 γ3 γ 1 γ 1 γ8 γ 4β + 41 β γ β + 1 β 1 β 1 γ8 γ 4β + 41 β c 4 γ β 1 1 γ8 γ 4β + 41 β + 1 γ1 β c γ β 1 =ϕx, say with x = c 1 Assume that ϕ attains maximum value in an interior point of x 0 [0, ]. A calculation shows that the maximum value occurs at 1 γ γ β x 0 = 1 β 1 γ8 γ 4β + 41 β 1 which is a contradiction. Thus, the maximum is attained on either x 0 = 0 or x 0 =. At x 0 = 0, we have ϕx 0 = ϕ0 =1 γ3 γ 1 γ 1 γ8 γ 4β + 41 β γ β + 1 β and at x 0 =, we have ϕx 0 = ϕ =ϕ0 1 β 1 γ8 γ 4β + 41 β γ β + 41 γ1 β 1 1 γ8 γ 4β + 41 β γ β. Now since and 1 1 γ8 γ 4β + 41 β>0 1 γ8 γ 4β + 41 β γ β <0
194 Goh Jiun Shyan and Aini Janteng where 0 β<1 and 0 γ<1, it is obvious that ϕ ϕ0. Hence ϕ attains maximum value at x 0 = 0. Finally, 9α + 1 a 3 4α + 1 1 γ8 γ 4β + 41 β a 9α + 1 γ β ϕ0. But ϕ0 1 γ + 1 β. Therefore, we get the result 9α+1 a 3 4α + 1 1 γ8 γ 4β + 41 β a 9α + 1 γ β 1 γ+1 β. When we substitute µ = 8α+1 9α+1 into 16, we manage to get the following: 9α + 1 a 8α + 1 3 9α + 1 a 1 γ + 1 β. 17 Hence completes the proof for the particular case: 8α + 1 9α + 1 µ 4α + 1 1 γ8 γ 4β + 41 β 9α + 1 γ β. The result is sharp by choosing c 1 = p 1 = 0, p = c =, b = 0, b 3 = 1 γ in 13. Finally, we consider the case µ 4α+1 1 γ8 γ 4β+41 β 9α+1 γ β. Write a 3 µa =a 3 4α + 1 1 γ8 γ 4β + 41 β a 9α + 1 γ β + { } 4α + 1 1 γ8 γ 4β + 41 β µ 9α + 1 γ β a
Fekete Szegö theorem 195 and thus 9α + 1α + 1 a3 µa 9α + 1α + 1 a 3 4α + 1 1 γ8 γ 4β + 41 β a 9α + 1 γ β + 9α + 1α + 1 µ 4α + 1 1 γ8 γ 4β + 41 β 9α + 1 γ β γ β 4α + 1 = α + 1 1 γ7 γ 4β + 1 β + 9α + 1 γ β µ 4 which results for µ = 4α+1 1 γ8 γ 4β+41 β 9α+1 γ β and the inequality a γ β α+1 have been used. By choosing c 1 = p 1 = i, c = p =, b = i1 γ, b 3 = 1 γ 1 γ in 13, the result is sharp. ACKNOWLEDGEMENTS. This work was supported by FRG068-ST- /010 Grant, Malaysia. The authors express their gratitude to the referee for his valuable comments. References [1] A. Janteng, S.A. Halim and M. Darus, Fekete-Szegö problem for certain subclass of quasi-convex functions, Int. J. Contemp. Math. Sci., 1 006, 45-51. [] B.A. Frasin and M. Darus, On the Fekete-Szegö using the Hadamard product, Ubter, Nath. J., 3 003, 197-1304. [3] Ch. Pommerenke, Univalent Functions, Göttingen, Vandenhoeck and Ruprecht, 1975. [4] F.R. Keogh and E.P. Merkes, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc., Proc. Amer. Math. Soc., 0 1969, 8-1. [5] J.S. Goh and A. Janteng, Fekete-Szegö functional for a subclass of quasiconvex functions, submitted. [6] M. Darus and D.K. Thomas, On the Fekete-Szegö theorem for close-toconvex functions, Math. Japonica, 44 1996, 507-511.
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