THE FEKETE-SZEGÖ COEFFICIENT FUNCTIONAL FOR TRANSFORMS OF ANALYTIC FUNCTIONS. Communicated by Mohammad Sal Moslehian. 1.

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1 Bulletin of the Iranian Mathematical Society Vol. 35 No. (009 ), pp THE FEKETE-SZEGÖ COEFFICIENT FUNCTIONAL FOR TRANSFORMS OF ANALYTIC FUNCTIONS R.M. ALI, S.K. LEE, V. RAVICHANDRAN AND S. SUPRAMANIAM Communicated by Mohammad Sal Moslehian Abstract. Sharp bounds for the Fekete-Szegö coefficient functional associated with the k-th root transform [f(z k )] 1/k of normalized analytic functions f(z) defined on the open unit disk in the complex plane are derived. A similar problem is investigated for functions z/f(z) when f belongs to a certain class of functions. 1. Introduction Let A be the class of all functions f(z) analytic in the open unit disk := {z C : z < 1}, normalized by f(0) = 0 f (0) = 1. For a univalent function in the class A, it is well known that the n-th coefficient is bounded by n. The bounds for the coefficients give information about the geometric properties of these functions. For example, the bound for the second coefficient of normalized univalent functions readily yields the growth distortion bounds for univalent functions. The Fekete- Szegö coefficient functional also arises naturally in the investigation of univalency of analytic functions. Several authors have investigated the Fekete-Szegö functional for functions in various subclasses of univalent MSC(000): Primary: 30C45, 30C50; Secondary: 30C80. Keywords: Analytic functions, starlike functions, convex functions, p-valent functions, subordination, Fekete-Szegö inequalities. Received: 5 June 008, Accepted: 7 November 008 Corresponding author. c 009 Iranian Mathematical Society. 119

2 10 Ali, Lee, Ravichran Shamani multivalent functions [1 5, 7 11, 14, 15, 19, 0, ], more recently by Choi et al. [6]. Recall that a function f A is starlike if f( ) is a starlike domain convex if f( ) is a convex domain. The classes consisting of starlike convex functions are usually denoted by S C, respectively. The function f(z) is subordinate to the function g(z), written f(z) g(z), provided that there is an analytic function w(z) defined on with w(0) = 0 w(z) < 1 such that f(z) = g(w(z)). A function f A is starlike if only if Re zf (z)/f(z) > 0, or equivalently if zf (z)/f(z) (1 + z)/(1 z). The superordinate function φ(z) = (1 + z)/(1 z) is a convex function. Ma Minda [13] have given a unified treatment of various subclasses consisting of starlike convex functions for which either one of the quantities zf (z)/f(z) or 1+zf (z)/f (z) is subordinate to a more general superordinate function. In fact, they considered the analytic function ϕ with positive real part in the unit disk, ϕ(0) = 1, ϕ (0) > 0, where ϕ maps onto a region starlike with respect to 1 symmetric with respect to the real axis. The unified class S (ϕ) introduced by Ma Minda [13] consists of functions f A satisfying zf (z) f(z) ϕ(z), (z ). They also investigated the corresponding class C(ϕ) of convex functions f A satisfying 1 + zf (z) f (z) ϕ(z). Ma Minda [13] obtained subordination results, distortion, growth rotation theorems. They also obtained estimates for the first few coefficients determined bounds for the associated Fekete-Szegö functional. A function f S (ϕ) is said to be starlike function with respect to ϕ, a function f C(ϕ) is a convex function with respect to ϕ. Several authors [1, 16 18, 1] have studied the classes of analytic functions defined by using the expression (zf (z))/f(z)+α(z f (z))/f(z). The unified treatment of various subclasses of starlike convex functions by Ma Minda [13] motivates one to consider similar classes defined by subordination. Here, we consider the following classes of

3 Fekete-Szegö coefficient functional 11 functions, { f A : ( f (z) 1 ) } ϕ(z), b R b (ϕ) := { S (α, ϕ) := f A : zf (z) f(z) + f (z) αz f(z) { ( zf (z) L(α, ϕ) := f A : f(z) M(α, ϕ) := } ϕ(z), ) α ( 1 + zf (z) f (z) { f A : (1 α) zf (z) f(z) + α ) 1 α ϕ(z)} ( 1 + zf (z) f (z) ), } ϕ(z), where z, b C \ {0} α 0. Functions in the class L(α, ϕ) are called logarithmic α-convex functions with respect to ϕ the functions in the class M(α, ϕ) are called α-convex functions with respect to ϕ. Some coefficient problems for functions f belonging to certain classes of p-valent functions were investigated in [4]. For a univalent function f(z) of the form (1.1) f(z) = z + the k-th root transform is defined by: a n z n, n= (1.) F (z) := [f(z k )] 1/k = z + b kn+1 z kn+1. In Section, sharp bounds for the Fekete-Szegö coefficient functional b k+1 µb k+1 associated with the k-th root transform of the function f belonging to the above mentioned classes are derived. In Section 3, a similar problem is investigated for functions G, where G(z) := z/f(z) the function f belongs to the above mentioned classes. Let Ω be the class of analytic functions w, normalized by w(0) = 0, satisfying the condition w(z) < 1. The following two lemmas regarding the coefficients of functions in Ω are needed to prove our main results. Lemma 1.1 is a reformulation of the corresponding result for functions with positive real part due to Ma Minda [13]. n=1

4 1 Ali, Lee, Ravichran Shamani Lemma 1.1. [4] If w Ω (1.3) w(z) := w 1 z + w z + (z ), then, t if t 1 w tw1 1 if 1 t 1 t if t 1. For t < 1 or t > 1, equality holds if only if w(z) = z or one of its rotations. For 1 < t < 1, equality holds if only if w(z) = z or one of its rotations. Equality holds for t = 1 if only if w(z) = (0 λ 1) or one of its rotations, while for t = 1, equality holds if only if w(z) = z λ+z 1+λz (0 λ 1) or one of its rotations. z λ+z 1+λz Lemma 1.. [10] If w Ω, then, w tw 1 max{1; t }, for any complex number t. The result is sharp for the function w(z) = z or w(z) = z.. Coefficient bounds for the k-th root transformation In the first theorem below, the bound for the coefficient functional b k+1 µb k+1 corresponding to the k-th root transformation of starlike functions with respect to ϕ is given. Notice that the classes S (α, ϕ), L(α, ϕ) M(α, ϕ) reduce to the class S (ϕ) for appropriate choice of the parameters. Although Theorem.1 is contained in the corresponding results for the classes S (α, ϕ), L(α, ϕ) M(α, ϕ), it is stated proved separately here because of its importance in its own right as well as to illustrate the main ideas. Theorem.1. Let ϕ(z) = 1 + z + B z + B 3 z 3 +, σ 1 := 1 [ ( ) ] k B 1 + 1, σ := 1 [ ( ) ] k B

5 Fekete-Szegö coefficient functional 13 If f given by (1.1) belongs to S (ϕ), F is the k-th root transformation of f given by (1.), then, B1 k bk+1 µb (1 µ) + B k, if µ σ 1, B k+1 1 k, if σ 1 µ σ, for µ complex, b k+1 µb k+1 B 1 k (1 µ) B k, if µ σ, k max { 1; k (1 µ) + B }. Proof. If f S (ϕ), then there is an analytic function w Ω of the form (1.3) such that (.1) Since zf (z) f(z) = ϕ(w(z)). zf (z) f(z) = 1 + a z + ( a + a 3 )z + (3a 4 3a a 3 + a 3 )z ϕ(w(z)) = 1 + w 1 z + ( w + B w 1)z +..., then it follows from (.1) that (.) a = w 1 (.3) a 3 = 1 [w + (B + B 1)w 1]. For a function f given by (1.1), a computation shows that (.4) [f(z k )] 1/k = z + 1 ( 1 k a z k+1 + k a 3 1 ) k 1 k a z k+1 +. The equations (1.) (.4) yield: (.5) b k+1 = 1 k a,

6 14 Ali, Lee, Ravichran Shamani (.6) b k+1 = 1 k a 3 1 k 1 k a. By using (.) (.3) in (.5) (.6), it follows: hence, b k+1 = w 1 k b k+1 = 1 [ ] w + B w1 + B 1 w 1, k k b k+1 µb k+1 = k { [ w k (1 µ) B ] } w1. The first result is established by an application of Lemma 1.1. If k (1 µ) B 1, then, µ 1 [ ( ) ] k B (µ σ 1 ), Lemma 1.1 gives: b k+1 µb k+1 B 1 k (1 µ) + B k. For 1 k (1 µ) B 1, we have, [ ( ) ] 1 k B µ 1 [ ( ) k B + 1 Lemma 1.1 yields: For k (1 µ) B µ 1 b k+1 µb k+1 k. 1, we have, [ ( ) k B + 1 ] + 1 ] + 1 (µ σ ), it follows from Lemma 1.1 that b k+1 µb k+1 B 1 k (1 µ) B k. (σ 1 µ σ ),

7 Fekete-Szegö coefficient functional 15 The second result follows by an application of Lemma 1.: [ bk+1 µb k+1 = k w k (1 µ) B ] w 1 B { 1 k max 1; k (1 µ) + B }. Remark.. (1) In view of the Alexer result that f C(ϕ) if only if zf S (ϕ), the estimate for b k+1 µb k+1 for a function in C(ϕ) can be obtained from the corresponding estimates in Theorem.1 for functions in S (ϕ). The details are omitted here. () For k = 1, the k-th root transformation of f reduces to the given function f itself. Thus, the estimate given in equation (.1) of Theorem.1 is an extension of the corresponding result for the Fekete-Szegö coefficient functional corresponding to functions starlike with respect to ϕ. Similar remarks apply to other results in this section. The well-known Noshiro-Warschawski theorem states that a function f A with positive derivative in is univalent. The class R b (ϕ) of functions defined in terms of the subordination of the derivative f is closely associated with the class of functions with positive real part. The bound for the Fekete-Szegö functional corresponding to the k-th root transformation of functions in R b (ϕ) is given in Theorem.3. Theorem.3. Let ϕ(z) = 1 + z + B z + B 3 z 3 +. If f given by (1.1) belongs to R b (ϕ), F is the k-th root transformation of f given by (1.), then, b k+1 µb k+1 b 3k { max 1; 3b 4 ( 1 1 k + µ ) k B }. Proof. If f R b (ϕ), then there is an analytic function w(z) = w 1 z + w z + Ω such that (.7) b ( f (z) 1 ) = ϕ(w(z)).

8 16 Ali, Lee, Ravichran Shamani Since b ( f (z) 1 ) = 1 + b a z + 3 b a 3z +... ϕ(w(z)) = 1 + w 1 z + ( w + B w 1)z +..., then it follows from (.7) that (.8) a = bw 1 (.9) a 3 = b 3 (w + B w 1). By using (.8) (.9) in (.5) (.6), it follows: hence, b k+1 = bw 3k (.10) b k+1 µb k+1 = b 3k b k+1 = bw 1 k + bb w 1 3k b B 1 w 1 8k + b B 1 w 1 8k, { [ ( 3bB1 1 w 4 1 k + µ ) B ] } w1. k Applying Lemma 1. yields: bk+1 µb k+1 = b B [ ( 1 3k w 3bB k + µ ) B ] w 1 k b B { ( 1 3k max 1; 3b k + µ ) B } k. Remark.4. When k = 1 ϕ(z) = 1 + Az 1 + Bz ( 1 B A 1), Theorem.3 reduces to a result in [9, Theorem 4, p. 894].

9 Fekete-Szegö coefficient functional 17 It is important to consider the special case of functions f A having positive derivatives, in particular, functions f satisfying the subordination f (z) ϕ(z). The class of such functions is the special case of the class R b (ϕ), when b = 1. In fact, when b = 1, equation (.10) becomes: b k+1 µb k+1 = B { [ ( 1 3B1 1 w 3k 4 1 k + µ ) B ] } w1. k Lemma 1.1 now yields the following result. Corollary.5. If f A satisfies f (z) ϕ(z), then, bk+1 µb k+1 where, B 1 4k ( 1 1 k + µ ) + B k 3k, if µ σ 1, 3k, if σ 1 µ σ, B1 ( 1 4k 1 k + µ ) B k 3k, if µ σ σ 1 := 4kB 3B 1 4k 3 k + 1 σ := 4kB 3B 1 + 4k 3 k + 1. The following result gives the bounds for the Fekete-Szegö coefficient functional corresponding to the k-th root transformation of functions in the class S (α, ϕ). Theorem.6. Let ϕ(z) = 1 + z + B z + B 3 z 3 +. Define σ 1, σ υ by: [ k(1 + α) σ 1 := (1 + 3α) (1 + α) + B B ] 1(k 1)(1 + 3α) (1 + α) 1, [ k(1 + α) σ := (1 + 3α) (1 + α) + B B ] 1(k 1)(1 + 3α) (1 + α) + 1, υ := [ ] (k 1)(1 + 3α) µ(1 + 3α) + (1 + α) k(1 + α) k(1 + α) 1 B.

10 18 Ali, Lee, Ravichran Shamani If f given by (1.1) belongs to S (α, ϕ), F is k-th root transformation of f given by (1.), then k(1 + 3α) υ, if µ σ 1, b k+1 µb k+1 k(1 + 3α), if σ 1 µ σ, for µ complex, b k+1 µb k+1 k(1 + 3α) υ, if µ σ, max {1; υ }. k(1 + 3α) Proof. If f S (α, ϕ), then there is an analytic function w(z) = w 1 z + w z + Ω such that Since (.11) (.1) zf (z) f(z) + αz f (z) = ϕ(w(z)). f(z) zf (z) f(z) = 1 + a z + ( a + a 3 )z + αz f (z) f(z) = a αz (a α 6a 3 α)z, then equations (.11) (.1) yield: (.13) zf (z) f(z) + αz f (z) = 1+a (1+α)z +[(1+3α)a 3 (1+α)a f(z) ]z +. Since ϕ(w(z)) = 1 + w 1 z + ( w + B w 1)z +..., then equation (.13) gives: (.14) a = w 1 (1 + α) (.15) a 3 = [ ] 1 w + B w1 + B 1 w 1. (1 + 3α) (1 + α)

11 Fekete-Szegö coefficient functional 19 Using (.14) (.15) in (.5) (.6), we get b k+1 = b k+1 = 1 k w 1 (1 + α), ( ) 1 w + B w1 + B 1 w 1 B1 w 1 k(1 + 3α) (1 + α) k (k 1), (1 + α) hence, { [ ( b k+1 µb k+1 = (k 1)(1 + 3α) w k(1 + 3α) (1 + α) k(1 + α) ) µ(1 + 3α) + k(1 + α) 1 B ] } w1. The first part of the result is established by applying Lemma 1.1. If µ σ 1, then υ 1 hence Lemma 1.1 yields: bk+1 µb k+1 k(1 + 3α) υ. If σ 1 µ σ, then 1 υ 1 hence Lemma 1.1 yields: b k+1 µb k+1 k(1 + 3α). For µ σ, then υ 1 hence Lemma 1.1 yields b k+1 µb k+1 k(1 + 3α) υ. The second result follows by an application of Lemma 1.. Observe that Theorem.6 reduces to Theorem.1 when α = 0. Theorem.7. Let ϕ(z) = 1+ z +B z +B 3 z 3 + β = (1 α). Also, let [ ( ) k (α + β) B σ 1 := 1 (α + β) 3(α + 4β) (α + 3β) (k 1)(α + 3β) k ],

12 130 Ali, Lee, Ravichran Shamani σ := υ := [ k (α + β) (α + 3β) ], ( ) B + 1 (α + β) 3(α + 4β) (k 1)(α + 3β) k [ (α + β) 3(α + 4β) (α + β) + 1 (k 1)(α + 3β) k + µ k (α + 3β) ] B. If f given by (1.1) belongs to L(α, ϕ), F is k-th root transformation of f given by (1.), then, k(α + 3β) υ, if µ σ 1, (.16) b k+1 µb k+1 k(α + 3β), if σ 1 µ σ, for µ complex, bk+1 µb k+1 k(α + 3β) υ, if µ σ, k(α + 3β) max {1; υ }. Proof. If f L(α, ϕ), then there is an analytic function w(z) = w 1 z + w z + Ω such that ( zf ) (z) α ( (.17) 1 + zf ) (z) β f(z) f = ϕ(w(z)). (z) We have, zf (z) f(z) = 1 + a z + (a 3 a )z + (3a 4 + a 3 3a 3 a )z 3 +, therefore, ( zf ) (z) α ( (.18) = 1 + αa z + f(z) Similarly, αa 3 + α 3α a 1 + zf (z) f (z) = 1 + a z + (6a 3 4a )z +, ) z +.

13 Fekete-Szegö coefficient functional 131 therefore, ( (.19) 1 + zf ) (z) β f = 1 + βa z + ( 6βa 3 + (β 3β)a (z) ) z +. Thus, from (.18) (.19), ( zf ) (z) α ( 1 + zf ) (z) β f(z) f = 1 + (α + β)a z + [(α + 3β)a 3 (z) + (α + ] β) 3(α + 4β) a z +. Since ϕ(w(z)) = 1 + w 1 z + ( w + B w1)z +..., then it follows from (.17) that (.0) a = w 1 (α + β) (.1) a 3 = w + B w 1 (α + 3β) [(α + β) 3(α + 4β)]B1 w 1 4(α + β). (α + 3β) Using (.0) (.1) in (.5) (.6), we get b k+1 = hence, where, σ := (α + β) b k+1 = 1 k w 1 k(α + β), w k(α + 3β) + B w1 k(α + 3β) [(α + β) 3(α + 4β)]B1 w 1 4k(α + β) (α + 3β) B 1 w 1 (k 1) k (α + β), b k+1 µb k+1 = { w σw } 1, k(α + 3β) [ (α + β) 3(α + 4β) + 1 k (k 1)(α + 3β) + µ ] k (α + 3β) B. The results now follow by using lemmas

14 13 Ali, Lee, Ravichran Shamani Remark.8. We note that if k = 1, then inequality (.16) is the result established in [19, Theorem.1, p. 3]. For the class M(α, ϕ), we now get the following coefficient bounds. Theorem.9. Let ϕ(z) = 1 + z + B z + B 3 z 3 +. Also, let [ ( ) ] k (1 + α) B σ 1 := 1 + (1 + 3α) (k 1)(1 + α), (1 + α) [ ( ) ] k (1 + α) B σ := (1 + 3α) (k 1)(1 + α), (1 + α) υ := [(k (1 + α) 1)(1 + α) + µk ] (1 + α) (1 + 3α) B. If f given by (1.1) belongs to M(α, ϕ), F is k-th root transformation of f given by (1.), then, k(1 + α) υ, if µ σ 1, b k+1 µb k+1 k(1 + α), if σ 1 µ σ, for µ complex, b k+1 µb k+1 k(1 + α) υ, if µ σ, max {1; υ }. k(1 + α) Proof. If f M(α, ϕ), then there is an analytic function w(z) = w 1 z + w z + Ω such that (.) (1 α) zf ( (z) f(z) + α 1 + zf ) (z) f = ϕ(w(z)). (z) Since (.3) (1 α) zf (z) f(z) = (1 α)+a (1 α)z +(1 α)( a +a 3 )z + (.4) α ( 1 + zf ) (z) f = α + a αz + α(3a 3 a (z) )z,

15 Fekete-Szegö coefficient functional 133 then from equations (.3) (.4), it follows: (1 α) zf ( (z) f(z) + α 1 + zf ) (z) f = 1 + (1 + α)a z (z) +[ (1 + 3α)a + (1 + α)a 3 ]z +. Since ϕ(w(z)) = 1 + w 1 z + ( w + B w1)z +, then it follows from equation (.) that (.5) a = w 1 (1 + α) (.6) a 3 = [ 1 w + B w1 + (1 + ] 3α)B 1 w 1 (1 + α) (1 + α). By using (.5) (.6) in (.5) (.6), it follows: b k+1 = hence, where, σ := b k+1 = w 1 k(1 + α), [ 1 w + B w1 + (1 + ] 3α)B 1 w 1 k(1 + α) (1 + α) b k+1 µb k+1 = k(1 + α) {w σw 1}, B 1 w 1 (k 1) k (1 + α), [(k (1 + α) 1)(1 + α) + µk ] (1 + α) (1 + 3α) B. The results follow from lemmas we have b k+1 µb k+1 γυ. For σ 1 µ σ, then 1 υ 1 hence by Lemma 1.1, we have b k+1 µb k+1 γ. For µ σ, then υ 1 hence by Lemma 1.1, we have b k+1 µb k+1 γυ. Remark.10. When k = 1 α = 1, Theorem.9 reduces to a result in [13, Theorem 3, p. 164].

16 134 Ali, Lee, Ravichran Shamani 3. The Fekete-Szegö functional associated with z/f(z) Here, bounds for the Fekete-Szegö coefficient functional associated with the function G defined by (3.1) G(z) = z f(z) = 1 + d n z n, where f belongs to one of the classes S (ϕ), R b (ϕ), S (α, ϕ), L(α, ϕ) M(α, ϕ), are investigated. Proofs of the results obtained here are similar to those given in Section, hence the details are omitted. The following result is for functions belonging to the class S (ϕ). n=1 Theorem 3.1. Let ϕ(z) = 1 + z + B z + B 3 z 3 +, σ 1 := 1 B 3 1 B B1, σ := 1 + B1 3 B B1. If f given by (1.1) belongs to S (ϕ), G is a function given by (3.1), then, 1 4 B 1 4 B3 1(µ 1), if µ σ 1, d µd 1 1, if σ 1 µ σ, 1 4 B B3 1(µ 1), if µ σ, for µ complex, d µd 1 1 { max 1; (1 µ)b 1 1 } (B + B1). Proof. A computation yields: z (3.) f(z) = 1 a z + (a a 3 )z +. Using (3.1), (3.) yields: (3.3) d 1 = a (3.4) d = a a 3.

17 Fekete-Szegö coefficient functional 135 By using (.) (.3) in (3.3) (3.4), it follows: hence, d µd 1 = 1 d 1 = w 1 d = B 1w 1 1 [w + (B + B 1)w 1], { [ w (1 µ)b1 1 ] } (B + B1) w1. The result is established from an application of Lemma 1.1. The second result follows from Lemma 1.: d µd 1 = 1 [ w (1 µ)b1 1 ] (B + B1) w1 1 { max 1; (1 µ)b 1 1 } (B + B1). For the class R b (ϕ), the following coefficient bound is obtained. Theorem 3.. Let ϕ(z) = 1 + z + B z + B 3 z 3 +. If f given by (1.1) belongs to R b (ϕ), G is a function given by (3.1), then, d µd 1 b B { 1 max 1; (1 µ)b B }. Proof. Using (.8) (.9) in (3.3) (3.4), it follows: d 1 = bw 1 d = 1 4 b B 1w 1 b 3 (w + B w 1), hence, d µd 1 = b 3 { [ 3 w 4 (1 µ)b B ] } w1.

18 136 Ali, Lee, Ravichran Shamani Lemma 1. gives: d µd 1 = b B [ w 4 (1 µ)b B ] w 1 b B { 1 3k max 1; 3 4 (1 µ)b B }. For functions with positive derivative, the above theorem turns in to the following result. Corollary 3.3. If f A satisfies f (z) ϕ(z), then 1 4 (1 µ)b 1 B 3, if µ σ 1, d µd B 1 1 3, if σ 1 µ σ, 1 4 (1 µ)b 1 + B 3, if µ σ where, σ 1 := B 3B 1 σ := B. 3 3B 1 The following result gives the coefficient bounds for the class S (α, ϕ). Theorem 3.4. Let ϕ(z) = 1 + z + B z + B 3 z 3 +. Define σ 1, σ, υ γ by: [ σ 1 := (1 + α) 1 + (1 + α)b (1 + 3α)B1 [ σ := (1 + α) 1 + (1 + α)b (1 + 3α)B1 + + ] 1 (1 + α), (1 + 3α) (1 + 3α) ] 1 (1 + α) +, (1 + 3α) (1 + 3α) υ := (1 + 3α) (1 + α) µ(1 + 3α) (1 + α) + (1 + α) + B, γ := (1 + 3α).

19 Fekete-Szegö coefficient functional 137 If f given by (1.1) belongs to S (α, ϕ), G is a function given by (3.1), then, (1 + 3α) υ, if µ σ 1, d µd 1 (1 + 3α), if σ 1 µ σ, for µ complex, d µd 1 (1 + 3α) υ, if µ σ, max {1; υ }. (1 + 3α) Proof. By using the relations (.14) (.15) in (3.3) (3.4), it follows: d 1 = w 1 (1 + α), ( ) d = B 1 w 1 (1 + α) 1 w + B w1 + B 1 w 1, (1 + 3α) (1 + α) hence, { [ d µd B1 (1 + 3α) 1 = w (1 + 3α) (1 + α) µ(1 + 3α) (1 + α) + (1 + α) + B ] } w1. The result is established by an application of Lemma 1.1. The second result follows by an application of Lemma 1.: d µd 1 = (1 + 3α) + (1 + α) + B [ w B1 (1 + 3α) (1 + α) µ(1 + 3α) (1 + α) ] w1 (1 + 3α) max {1; υ }. For the class L(α, ϕ), we now get the following coefficient bounds.

20 138 Ali, Lee, Ravichran Shamani Theorem 3.5. Let ϕ(z) = 1+ z +B z +B 3 z 3 + β = (1 α). Also, let σ 1 := 1 (α + β) B (α + 3β)B 1 σ := 1 (α + β) B (α + 3β)B 1 + [(α + β) 3(α + 4β)](α + β) 4(α + 3β) + [(α + β) 3(α + 4β)](α + β) 4(α + 3β) 1, + 1, υ := (α + β) B + [(α + β) 3(α + 4β)] (α + β) µ. (α + 3β) 4(α + 3β) If f given by (1.1) belongs to L(α, ϕ), G is a function given by (3.1), then, υ (α + β), if µ σ 1, d µd 1 (α + β), if σ 1 µ σ, for µ complex, υ (α + β), if µ σ, d µd 1 max {1; υ }. (α + β) Proof. Using (.0) (.1) in (3.3) (3.4), it follows: d 1 = w 1 (α + β), d = B 1 w 1 (α + β) B w 1 + B w1 (α + 3β) Hence, where, + [(α + β) 3(α + 4β)]B1 w 1 4(α + β). (α + 3β) d µd { 1 = w (α + β) σw1 }, σ := (α + β) B + [(α + β) 3(α + 4β)] (α + β) µ. (α + 3β) 4(α + 3β)

21 Fekete-Szegö coefficient functional 139 The result now follows from Lemma 1.1. The second result follows by an application of Lemma 1.: d µd 1 = (α + β) w σw1 (α + β) max {1; σ }. Finally, for the class M(α, ϕ), the following coefficient bounds are obtained. Theorem 3.6. Let ϕ(z) = 1 + z + B z + B 3 z 3 +. Also, let σ 1 := 1 B (1 + α) (1 + α) B1 (1 + 3α) (1 + α) (1 + α), σ := 1 B (1 + α) (1 + α) B1 (1 + 3α) + (1 + α) (1 + α), υ := (1 + α) (1 + α) B (1 + α)(1 + 3α) (1 + α) µ(1 + α) (1 + α). If f given by (1.1) belongs to M(α, ϕ), G is a function given by (3.1), then, (1 + α) υ, if µ σ 1, d µd 1 (1 + α), if σ 1 µ σ, for µ complex, (1 + α) υ, if µ σ, d µd 1 max {1; υ }. (1 + α) Proof. Using (.5) (.6) in (3.3) (3.4), it follows: d 1 = w 1 (1 + α),

22 140 Ali, Lee, Ravichran Shamani [ d = B 1 w 1 (1 + α) 1 w + B w1 + (1 + ] 3α)B 1 w 1 (1 + α) (1 + α), hence, where, d µd { 1 = w (1 + α) σw1 }, σ := (1 + α) (1 + α) B (1 + α)(1 + 3α) (1 + α) µ(1 + α) (1 + α). The results follows from Lemma 1.1. The second result follows by an application of Lemma 1.: d µd 1 = (1 + α) w σw1 (1 + α) max {1; σ }. Acknowledgments The work presented here was supported in part by research grants from Universiti Sains Malaysia (Research University Grant), FRGS University of Delhi, was completed during the third author s visit to USM. References [1] S. Abdul Halim, On a class of functions of complex order, Tamkang J. Math. 30 () (1999) [] O. P. Ahuja M. Jahangiri, Fekete-Szegö problem for a unified class of analytic functions, Panamer. Math. J. 7 () (1997) [3] R. M. Ali, Starlikeness associated with parabolic regions, Int. J. Math. Math. Sci. 4 (005) [4] R. M. Ali, V. Ravichran N. Seenivasagan, Coefficient bounds for p-valent functions, Appl. Math. Comput. 187 (1) (007) [5] N. E. Cho S. Owa, On the Fekete-Szegö problem for strongly α-logarithmic quasiconvex functions, Southeast Asian Bull. Math. 8 (3) (004)

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24 14 Ali, Lee, Ravichran Shamani Rosihan M. Ali School of Mathematical Sciences, Universiti Sains Malaysia, USM, Penang, Malaysia See Keong Lee School of Mathematical Sciences, Universiti Sains Malaysia, USM, Penang, Malaysia V. Ravichran Department of Mathematics, University of Delhi, Delhi , India Shamani Supramaniam School of Mathematical Sciences, Universiti Sains Malaysia, USM, Penang, Malaysia