Coefficient Bounds for Certain Subclasses of Analytic Function Involving Hadamard Products

Transcription

1 Mathematica Aeterna, Vol. 4, 2014, no. 4, Coefficient Bounds for Certain Subclasses of Analytic Function Involving Hadamard Products V.G. Shanthi Department of Mathematics S.D.N.B. Vaishnav College for Women Chromepet, Chennai , India vg.shanthi1@gmail.com B. Srutha Keerthi Mathematics Division, School of Advanced Sciences VIT University Chennai Campus Vandallur Kellambakkam Road, Chennai , India sruthilaya06@yahoo.co.in B. Adolf Stephen Department of Mathematics Madras Christian College, Tambaram Chennai adolfmcc2003@yahoo.co.in µa 2 2 Abstract In this present investigation authors obtained sharp boundsfor a 3 for certain subclasses of analytic functions when µ 1. Mathematics Subject Classification: 30C45 Keywords: Analytic functions, starlike functions, convex functions, univalent functions, subordination, convolution, coefficient inequalities. 1 Introduction Let A denote the family of functions of the form f(z) = z + a n z n, (1) n=2

2 404 V.G. Shanthi, B. Srutha Keerthi and B. Adolf Stephen which are analytic in the open unit disk := {z : z < 1}. Further, let S denote the class of functions which are univalent in. Let the function be defined by (1). Also let the function g(z) be defined by g(z) = z + b n z n. (2) Then the Hadamard product (or convolution) of the functions f(z) and g(z) is defined by f(z) g(z) = z + a n b n z n. (3) Fekete and Szegö [8] obtained sharp upper bounds for a 3 µa 2 2 when µ is real. For various subclasses of S, sharp upper bound for functional a 3 µa 2 2 has been studied by many different authors including [1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. In this paper we obtain sharp upper bounds for a 3 µa 2 2 when f(z) belonging to the class of functions defined as follows: Definition 1.1. Let 0 < α 1, 0 λ 1, and β > 0, and let f(z) A. Then f(z) M(Φ,Ψ,λ,α,β) if and only if [ ( )] (1 λzf arg (z) f (z)+zf (z) + < πα g(z) g (z) 2, (4) with g(z) A and satisfies ( ) g(z) Φ(z) arg < πβ,z (5) g(z) Ψ(z) 2 where Φ(z) = z + n=2 Υ nz n and Ψ(z) = z+ n=2 γ nz n, analytic in such that g(z) Ψ(z) 0, Υ n 0, γ n 0 and Υ n > γ n (n 2). Definition 1.2. Let 0 < α 1, 0 λ 1, β > 0, and let f(z) A. Then f(z) N(Φ,Ψ,λ,α,β) if and only if ( ) z (1 λ) arg f (z) < πα,z (6) g(z) (1 λ) 2 with g(z) A and satisfies equation (5). Lemma 1.3 ([18]). Let h(z) P that is h(z) be analytic in and be given by h(z) = 1+c 1 z +c 2 z 2 + and Reh(z) > 0 for z then c 2 c c (7) n=2 n=2

3 Coefficient Bounds for Certain Subclasses of Analytic Function Coefficient Problem Theorem 2.1. Let f(z) be given by (1). If f(z) M(Φ,Ψ,λ,α,β) and 3ηµ 2δ 2 +4δγ 2, where δ = Υ 2 γ 2, η = Υ 3 γ 3 and µ 1, then we have the sharp inquality a 3 µa 2 2 β2 3ηδ 2[3µη 2δ2 4δγ 2 ) + α[αδ(3µ(1+2λ) 2(1+λ2 ))+2β(1+λ)(3µ(1+2λ) 2(1+3λ))] 3δ(1+λ) 2 (1+2λ) Proof. Let f(z) M(Φ,Ψ,λ,α,β). It follows from (4) that (1 λ)zf (z)g (z)+λ[f (z)+zf (z)]g(z) (8) = q α (z)g(z)g (z) (9) for z, with q(z) P given by q(z) = 1+q 1 z+q 2 z 2 +q 3 z 3 +. Equating coefficients, we obtain and 2a 2 (1+λ) = b 2 (1+λ)+αq 1 (10) 1 3a 3 (1+2λ) =b 3 (1+2λ)+3αq 1 b 2 2αq 1 b 2 1+λ +αq 2 + α(α 1) q1 2 (11) 2 Also, it follows from (5) that g(z) Φ(z) = [g(z) Ψ(z)]p β (z), (12) where p(z) P with p(z) = 1 + p 1 z + p 2 z 2 + for z. Thus equating coefficients, we obtain δb 2 = βp 1 (13) [ ηb 3 = β From (10), (11), (13), (14) we have a 3 µa 2 2 = α 3(1+2λ) p 2 + β(δ +2γ 2) δ p 2 1 2δ [ ] q 2 q2 1 + β [ ] p 2 p η 2 ]. (14) + α2 q 2 1 [2(1+λ)2 3µ(1+2λ)] 12(1+λ) 2 (1+2λ) + αβp 1q 1 [2(1+3λ) 3µ(1+2λ1)] 6δ(1+λ)(1+2λ) + β2 p 2 1(2δ 2 +4γ 2 δ 3µη) 12ηδ 2. (15)

4 406 V.G. Shanthi, B. Srutha Keerthi and B. Adolf Stephen Assume that a 3 µa 2 2 positive. Thus we now estimate Re(a 3 µa 2 2 ) byapplying the same technique done by London [15], and so from (15) and by using lemma 1.3 and letting p 1 = 2re iθ, q 1 = 2Re iφ, 0 r 1, 0 R 1, 0 θ 2π and 0 φ 2π, we obtain, Re[a 3 µa 2 α 2] = 3(1+2λ Re ( q 2 q2 1 )+ β2η ( ) 2 Re p 2 p α2 [2(1+λ) 2 3µ(1+2λ)]Re q (1+λ) 2 (1+2λ) + αβ[2(1+3λ) 3µ(1+2λ)]Re p 1q 1 6δ(1+λ)(1+2λ) + β2 (2δ 2 +4γ 2 δ 3µη)Re p ηδ 2 2α 3(1+2λ) (1 R2 )+ 2β 3η (1 r2 ) + α2 [2(1+λ) 2 3µ(1+2λ)]R 2 cos2φ 3(1+λ) 2 (1+2λ) + 2αβ[2(1+3λ) 3µ(1+2λ)]rRcos(θ+φ) 3δ(1+λ)(1+2λ) + β2 (2δ 2 +4γ 2 δ 3µη)r 2 cos2θ 3ηδ 2 2α 3(1+2λ) (1 R2 )+ 2β 3η (1 r2 ) + α2 [3µ(1+2λ) 2(1+λ) 2 ]R 2 3(1+λ) 2 (1+2λ) + 2αβ[3µ(1+2λ) 2(1+3λ)]rR 3δ(1+λ)(1+2λ) + β2 (3µη 2δ 2 4γ 2 δ)r 2 = G(r,R) 3ηδ 2 Letting α,β and µ fixed and differentiating G(r,R) partially when 0 < α 1,

5 Coefficient Bounds for Certain Subclasses of Analytic Function 407 β 1 and µ 1 we observe that G rr G RR (G rr ) 2 = 16αβ 9η(1+2λ) + 4α2 β 2 (3µη 2δ 2 4γ 2 δ)[3µ(1+2λ) 2(1+λ) 2 9ηδ 2 (1+λ) 2 (1+2λ) 8α2 β[3µ(1+2λ) 2(1+λ) 2 ] 9η(1+λ) 2 (1+2λ) 8αβ2 [3µη 2δ 2 4γ 2 δ] 9ηδ 2 (1+2λ) 4α2 β 2 [3µ(1+2λ) 2(1+3λ)] 2 9δ 2 (1+λ) 2 (1+2λ) 2 < 0. Therefore, the maximum of G(r, R) occurs on boundaries. Thus the desired inequality follows by observing that G(r,R) G(1,1) = β2 (3µη 2δ 2 4γ 2 δ) 3ηδ 2 + α[αδ(3µ(1+2λ) 2(1+λ)2 )+2β(1+λ)(3µ(1+2λ) 2(1+3λ))] 3δ(1+λ) 2 (1+2λ) (16) The equality is attained when choosing p 1 = q 1 = 2i and p 2 = q 2 = 2 in (15). Theorem 2.2. Let f(z) given by (1). If f(z) N(Φ,Ψ,λ,α,β) and 3ηµ 2δ 2 +4γ 2 δ where δ = Υ 2 γ 2, η = Υ 3 γ 3 and µ 1, then we have the sharp inequality a 3 µa 2 2 β2 (1 λ) 3ηδ 2 [3µη(1 λ) 2δ 2 4γ 2 δ] + α[αδ+2β(1 λ)](3µ 2) 3δ + 2λ(1 λ)β2 3δ 2 (17) Proof of Theorem 2.2 is similar to Theorem 2.1. So the details are omitted. Remark 2.3. Letting λ = 0 in Theorem 2.1, we have the result given by Darus and Thomas [7]. Remark 2.4. Letting Φ(z) = z (1 z) 2, Ψ(z) = z 1 z, λ = 0 and α = 1 in Theorem 2.1, we have the result given by Jahangiri [11].

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