Fekete-Szegö Inequality for Certain Classes of Analytic Functions Associated with Srivastava-Attiya Integral Operator

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1 Applied Mathematical Sciences, Vol. 9, 015, no. 68, HIKARI Ltd, Fekete-Szegö Inequality for Certain Classes of Analytic Functions Associated with Srivastava-Attiya Integral Operator C. Ramachandran Department of Mathematics University College of Engineering Villupuram Anna University Chennai, Villupuram , Tamilnadu, India K. Dhanalakshmi Department of Mathematics IFET College of Engineering Gangarampalayam, Villupuram , Tamilnadu, India L. Vanitha Department of Mathematics University College of Engineering Villupuram Anna University, Villupuram , Tamilnadu, India Copyright c 014 K. Dhanalakshmi, L. Vanitha and C. Ramachandran. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The aim of this paper is to establish the Fekete-Szegö Inequality for certain classes of analytic functions which is associated with Srivastava- Attiya integral operator. Certain applications of these results for the functions defined through convolution are also obtained. Mathematics Subject Classification: 30C45

2 3358 C. Ramachandran, K. Dhanalakshmi and L. Vanitha Keywords: Univalent function, Hadamard product, Star like function, Convex function, Fractional Calculus 1 Introduction Let A denote the class of analytic functions f(z) be of the form defined on the open unit disk f(z) = z + a k z k (1) k= = {z : z C : z < 1} Also let S be the subclass of A consisting of univalent functions in A functionf S is said to be starlike if, if f maps conformally onto the starlike domain with respect to the origin and the class of all starlike functions are denoted by S, analytically this class is ( characterized by the inequality,the ) zf (z) function f S is starlike if and only if Re > 0, for z. f(z) A function f S is said to be convex, if f maps conformally onto the convex domain with respect to the origin and the class of all convex functions are denoted by C, analytically this class is characterized by the inequality, the ) zf (z) function f S is convex, if and only if Re (1 + > 0, for z f (z) With a view to recalling the principal of subordination between analytic functions, let the functions f and g be analytic in. Then we say that the function f is subordinate to g, if there exits a Schwarz function ω, analytic in with ω(0) = 0 and ω(z) < 1 (z ) such that We denote this subordination by f(z) = g(ω(z)) (z ) f g or f(z) g(z). In particular, if the function g is univalent in, the above subordination is equivalent to f(0) = g(0) and f( ) g( ).

3 On the Fekete-Szegö inequality 3359 For two analytic functions f(z) = z + a n z n and g(z) = z + b n z n, their convolution (or Hardamard Product) is defined by the analytic function (f g)(z) = z + a n b n z n. z n= The classes of starlike and convex functions are closed under convolution with convex functions. This was conjectured by Pólya and Schoenberg 11] and proved by Ruscheweyh and Sheil-Small16]. Definition ] The general Hurwitz-Lerch zeta function Φ(a, s; z) is defined as Φ(a, s; z) = 1 a s + n= n= z () + z s ( + a) + = z k s (k + a), z, s a C\Z 0 = {0, 1,,... }, s C and for z = 1, Re s > 1. k=0 Definition 1.. For s C and a C\Z 0, Srivatsava and Attiya 14] defined the operator L a,s : A A as below L a,s f(z) = G a,s (z) f(z) = z + k= G a,s (z) = (1 + a) s ( Φ(a, s; z) 1 a s ) = ( ) s 1 + a a k z k, z. k + a k=1 ( ) s 1 + a z k. k + a The above operator is known as Srivastava and Attiya integral operator Many well-known subclasses of analytic functions are investigated by using the Srivatsava-Attiya Integral Operator 5, 17]. Definition ] Let λ > 1, a C\Z 0, and s C the operator D s a,λ can be defined as D s a,λf(z) = G ( 1) a,s f(z) = z + G ( 1) a,s (z) = z + k+ k= ( ) s k + a (k + λ 1)! a k z k. 1 + a λ!(k 1)! ( ) s k + a (k + λ 1)! z k, z. 1 + a λ!(k 1)! For different choices of parameters, the operatorda,λ s studied by various authors. was extensively

4 3360 C. Ramachandran, K. Dhanalakshmi and L. Vanitha Definition ] For s C, a C\Z 0, λ > 1, µ > 0, and operator (Da,λ s (z)) 1 is defined in terms of convolution as D s a,λ(z) (D s a,λ(z)) ( 1) = z (1 z) µ, z, The Definition ] Let Ia,λ,µ s : A A can be defined as ( F(z) = Ia,λ,µf(z) s = (Da,λ(z)) s ( 1) f(z) = z+ 1+a ) s (µ) k 1 λ! k+a (k + λ 1)! a kz k () (µ) k = k= { 1 if k = 0 µ(µ + 1)... (µ + k 1) if k = 1,,.... For a = 0, λ = 1, µ = and s = 0, we get, I 0 0,1,f(z) = f(z) and for a = 0, λ = 0, µ = and s = 0, we get, I 0 0,0,f(z) = zf (z) Let φ(z) be an analytic function with positive real part on with φ(0) = 1, φ (0) > 0 which maps the open unit disk onto a region starlike with respect to 1 which is symmetric with respect to the real axis. Let S (φ) be the class of all functions f S for which zf (z) f(z) φ(z), (z ) and C(φ) be the class of functions in f S for which 1 + zf (z) f (z) φ(z), (z ). These classes were investigated and studied by Ma and Minda6] Definition ] Let φ(z) = 1+B 1 z +B z + be a univalent starlike function with respect to 1 which maps the open unit disk onto a region in the right half plane which is symmetric with respect to the real axis, and let B 1 > 0. The function f S is in the class F s (φ) if zf (z) F(z) The function f S is in the class F c (φ) if 1 + φ(z) zf (z) F (z) φ(z)

5 On the Fekete-Szegö inequality 3361 In virtue of Löwners method, Fekete and Szegö 4] proved the striking result, if f S then 3 4ν if ν 0 a 3 νa 1 + exp ( ) ν if 0 ν 1 1 ν 4ν 3 if ν 1 In the present paper, the Fekete-Szegö inequality for the functions in the subclasses F s (φ) and F c (φ) of F(z) are obtained. For various functions of S, the upper bound a 3 νa is investigated by many different authors including 1,, 7, 1] In order to derive the main result the following lemma is required. Lemma ] If p(z) = 1 + c 1 z + c z +... is an analytic function with a positive real part, then 4η + if η 0 c ηc 1 if 0 η 1 4η if 1 When η < 0 or η > 1, the equality holds if and only if p(z) is 1 + z or one of 1 z its rotations. If 0 < η < 1, then the equality holds if and only if 1 + z or one 1 z of its rotations. Equality holds if and only if p(z) = ( ) ( 1 + z 1 λ 1 z + 1 ) 1 z λ 1 + z (0 λ 1) or one of its rotations. If η= 1, the equality holds if and only if p(z) is the reciprocal of one of the function such that the equality holds in the case of η = 0. Also the above upper bound is sharp, it can be improved as follows when 0 < η < 1: c ηc 1 + (η) c 1 (0 < η 1/) and c ηc 1 + (1 η) c 1 1 (1/ < η 1). Fekete-Szegö Inequality The following are the main result of the paper:

6 336 C. Ramachandran, K. Dhanalakshmi and L. Vanitha Theorem.1. Let φ(z) = 1 + B 1 z + B z + If f(z) given by (1) belongs to F s (φ) then, a 3 νa and σ 1 = 1 B 1 σ = 1 B 1 The result is sharp. (λ + 1)(λ + ) (µ) (λ + 1)(λ + ) (µ) (λ + 1)(λ + ) (µ) P roof. F orf F s, let From (3), we obtain ( ) s a + 3 λ + 1 B + B1 (µ) ν λ + ((µ) 1 ) B 1 if ν σ 1 B 1 if σ 1 ν σ ( a + 3 ) s B + B 1 ν λ + 1 (µ) λ + ((µ) 1 ) B 1 ( ) λ + ((µ)1 ) (a + 3) s () s λ + 1 (µ) (a + ) s B 1 + B1) ( ) λ + ((µ)1 ) (a + 3) s () s λ + 1 (µ) (a + ) s + B 1 + B1) p(z) = zf (z) F(z) = 1 + b 1z + b z +... (3) ( ) s a + (λ + 1)b 1 = a (µ) 1 and (λ + 1)(λ + ) ( ) b + b 1 = a3. (µ) Since φ(z) is univalent and p φ, the function p 1 (z) = 1 + φ 1 (p(z)) 1 φ 1 (p(z)) = 1 + c 1z + c z +... if ν σ is analytic and has positive real part in. Also, ( ) p1 (z) 1 p(z) = φ p 1 (z) + 1 (4) and from (4), b 1 = 1 B 1c 1

7 On the Fekete-Szegö inequality 3363 and Therefore a 3 νa = B 1 4 b = 1 B 1(c 1 c 1) B c 1. ( ) s 3 + a (λ + 1)(λ + ) (c ηc 1 + a (µ) 1) η = 1 1 B B 1 B 1 + νb 1 λ + 1 (µ) (a + ) s λ + ((µ) 1 ) () s (a + 3) s The result now follows by an application of Lemma 1.7 To show that these bounds are sharp, define the functions, K φn (n =, 3,... ) by zk φ n (z) K φn (z) = φ(zn 1 ), K φn (0) = 0 = K φn ] (0) 1 and the function F λ and G λ (0 λ 1) is given by zf λ (z) ( ) z(z + λ) F λ (z) = φ, F λ (0) = 0 = F λ ] (0) λz and zg λ (z) ( ) z(z + λ) G λ (z) = φ, G λ (0) = 0 = G λ ] (0) λz Clearly the functions K φn, F λ, G λ F s (φ). Also we write K φ := K φ. If ν < σ 1 or ν < σ, then the equality holds if and only if f is K φ or one of its rotations. When σ 1 < ν < σ, the equality holds if and only if f is K φ3 or one of its rotations. If ν = σ 1 then the equality holds if and only if f is F λ or one of its rotations. If ν = σ then the equality holds if and only if f is G λ or one of its rotations. If σ 1 ν σ, consequence of Lemma 1.7 and Theorem.1 we get the following result Theorem.. Let f(z) given by (1) belongs to F s (φ). Let σ 3 be given by If σ 1 ν σ 3, then, σ 3 = δ 1 ( B + B B 1) 1 a 3 νa + δ 1 B1 B B1 + β 1 B1] a (λ + 1)(λ + ) (µ) ( ) s 3 + a B 1

8 3364 C. Ramachandran, K. Dhanalakshmi and L. Vanitha If σ 3 ν σ, then, a 3 νa + δ 1 B1 + B B1 β 1 B1] a Where δ 1 = (λ + 1)(λ + ) (µ) ( ) λ + ((µ)1 ) () s (a + 3) s λ + 1 (µ) (a + ) s B 1 β 1 = ν(λ + 1)(µ) (a + ) s (λ + ) ((µ) 1 ) (3 + a) s () s (λ + ) ((µ) 1 ) (3 + a) s () s Theorem.3. If f(z) given by (1) belongs to F c (φ) then, δ B + B1 3 λ + 1 ν (µ) (a + ) s B1 if ν σ λ + ((µ) 1 ) () s (a + 3) s 1 a 3 νa δ B 1 if σ 1 ν σ δ B + B1 3 λ + 1 ν (µ) (a + ) s B1 if ν σ λ + ((µ) 1 ) () s (a + 3) s σ 1 = 3B 1 σ = 3B 1 (λ + 1)(λ + ) δ = 6(µ) ( ) λ + ((µ)1 ) (3 + a) s () s ( ) B B λ + 1 (µ) (a + ) s 1 + B1 ( ) λ + ((µ)1 ) (a + 3) s () s λ + 1 (µ) (a + ) s + B 1 + B1) The result is sharp. Remark.4. For the special case a = 0, λ = 1, µ = and s = 0, in Theorem.1and a = 0, λ = 0, µ = and s = 0 in Theorem.3 represent the results obtained by Ma and Minda6]. Theorem.5. If f(z) given by (1) belongs to F c (φ). Let σ 3 be given by If σ 1 µ σ 3, then, σ 3 = δ 1 ( B + B 3B 1) 1 a 3 νa + δ 1 B1 B 3B1 + β B1] a (λ + 1)(λ + ) 6(µ) B a

9 On the Fekete-Szegö inequality 3365 If σ 3 ν σ, then, a 3 νa + δ 1 B1 + B 3B1 β B1] a Where δ 1 = (λ + 1)(λ + ) 6(µ) ( ) λ + ((µ)1 ) () s (a + 3) s λ + 1 (µ) (a + ) s B 1 β = 3ν(λ + 1)(µ) (a + ) s (λ + ) ((µ) 1 ) (a + 3) s (1 + a) s (λ + ) ((µ) 1 ) (a + 3) s (1 + a) s 3 Applications to functions defined by fractional derivatives For a fixed g A, let F g s (φ) be the class of functions f A for which (f g) F s (φ) and F g c (φ) be the class of functions f A for which (f g) F c (φ). Definition , 10, 1] Let f(z) be analytic in a simply connected region of the z-plane containing the origin. The fractional derivative f of order λ is defined by D λ z f(z) = 1 d Γ(1 λ) dz z 0 f(ζ) dζ (0 λ < 1). (z ζ) λ the multiplicity of (z ζ) λ is removed by requiring that log(z ζ) is real for z ζ > 0. With the help of definition 3.1 and its known extensions involving fractional derivatives and fractional integrals, Owa and Srivatsava 10] introduced the operator Ω λ : A A defined by (Ω λ f)(z) = Γ( λ)z λ D λ z f(z), (λ, 3, 4... ). The classes F λ s (φ) and F λ c (φ) consist of functions f A for which Ω λ f F s (φ) and Ω λ f C s (φ) respectively. The class F λ s (φ) is the special case of the class F g s (φ) and the class F λ c (φ) is the special case of the class F g s (φ) when g(z) = z + n= Γ (n + 1) Γ ( λ) z n (5) Γ (n + 1 λ) Since f F g s (φ)), F g c (φ)) if and only if f g S s (φ)(c s (φ)), we obtain the co-efficient estimates for functions in the classes F g s (φ) and F g c (φ), from

10 3366 C. Ramachandran, K. Dhanalakshmi and L. Vanitha the corresponding estimates for functions in the classes F s (φ) and F c (φ). Applying Theorem.1 for the function (f g)(z) = z + g a z + g 3 a 3 z , we get the following theorem after choosing the suitable parameter µ : Theorem 3.. Let g(z) = z + g k z k (g n > 0). If f(z) given by (1) belongs to Fs g (φ) then, a 3 νa σ 1 = g B 1g 3 σ = g B 1g 3 The result is sharp. (λ + 1)(λ + ) g 3 (µ) (λ + 1)(λ + ) g 3 (µ) (λ + 1)(λ + ) g 3 (µ) k= ( ) s a + 3 B + B1 νg 3 λ + 1 (µ) if ν σ g λ + ((µ) 1 ) 1 B 1 if σ 1 ν σ ( a + 3 ) s B + B 1 νg 3 g ( ) λ + 1 (µ) ] λ + ((µ) 1 ) ( ) λ + ((µ)1 ) (a + 3) s () s λ + 1 (µ) (a + ) s B 1 + B1) ( ) λ + ((µ)1 ) (a + 3) s () s λ + 1 (µ) (a + ) s + B 1 + B1) Theorem 3.3. Let g(z) = z + g k z k (g n > 0). If f(z) given by (1) belongs to Fc g (φ) then, a 3 νa (λ + 1)(λ + ) 6(u) g 3 (λ + 1)(λ + ) 6(u) g 3 (λ + 1)(λ + ) 6(u) g 3 k= ] B + B1 δ 4 if ν σ 1 B 1 if σ 1 ν σ ( a + 3 ) s B + B 1 δ 4 ] if ν σ if ν σ σ 1 = g 3B 1g 3 δ 4 = 3νg ( 3 λ + 1 g λ + ( ) λ + ((µ)1 ) λ + 1 ) (µ) ((µ) 1 ) (a + ) s B 1 () s (a + 3) s (µ) (a + 3) s () s (a + ) s ( B B 1 + 4B 1 )

11 On the Fekete-Szegö inequality 3367 σ = g 3B 1g 3 The result is sharp. Since We have and ( ) λ + ((µ)1 ) (a + 3) s () s λ + 1 (µ) (a + ) s + B 1 + 4B1) ( Ω λ f ) (z) = z + g 3 := g := n= Γ(3)Γ( λ) Γ(3 λ) Γ(4)Γ( λ) Γ(4 λ) Γ (n + 1) Γ ( λ) a n z n Γ (n + 1 λ) = = λ combining (6) and (7), Theorem 3.3 is reduce as follows: (6) 6 ( λ)(3 λ), (7) Theorem 3.4. Let λ <. If f(z ) given by (1) belongs to Fs λ (φ) then, δ 5 B + B 3ν( λ) λ + 1 (µ) 1 if ν σ (3 λ) λ + ((µ) 1 ) 1 a 3 νa δ 5 B 1 if σ 1 ν σ δ 5 B + B1 3ν( λ) λ + 1 (µ) if ν σ (3 λ) λ + ((µ) 1 ) σ 1 = σ = δ 5 = (3 λ) 3B 1( λ) (3 λ) 3B 1( λ) (λ + 1)(λ + )( λ)(3 λ) 1(µ) ( ) λ + ((µ)1 ) λ + 1 (µ) (a + 3) s () s (a + ) s B 1 + B 1) ( ) λ + ((µ)1 ) (a + 3) s () s λ + 1 (µ) (a + ) s + B 1 + B1) Using the value of g and g 3, Theorem 3.4 is reduce as follows: Theorem 3.5. Let λ <. If f(z) given by (1) belongs to Fc λ (φ) then, δ 6 B + B 9ν( λ) λ + 1 (µ) (a + ) s B1 1 if ν σ 4(3 λ) λ + ((µ) 1 ) () s (a + 3) s 1 a 3 νa δ 6 B 1 if σ 1 ν σ δ 6 B + B1 9ν( λ) λ + 1 (µ) (a + ) s B1 if ν σ 4(3 λ) λ + ((µ) 1 ) () s (a + 3) s

12 3368 C. Ramachandran, K. Dhanalakshmi and L. Vanitha and σ 1 = σ = δ 6 = 4(3 λ) 9B 1( λ) 4(3 λ) 9B 1( λ) The result is sharp. References (λ + 1)(λ + )( λ)(3 λ) 36(µ) ( ) λ + ((µ)1 ) λ + 1 (µ) (a + 3) s () s (a + ) s ( B B 1 + B 1 ( ) λ + ((µ)1 ) (a + 3) s () s λ + 1 (µ) (a + ) s + B 1 + B1) 1] OM. Ahuja and H. Orhan, The Fekete-Szegö problem for a class of analytic functions defined by Carlson-Shaffer operator, Ann. Univ. Mariae Curie-Sk lodowska Sect. A 68 (014), no. 1, ] S. Annamalai, C. Ramachandran and G. Murugusundaramoorthy, Fekete Szegö coefficient for the Janowski α-spirallike functions in open unit disk, Int. J. Math. Anal. (Ruse) 8 (014), no. 17-0, ] N. E. Cho and J. A. Kim, Inclusion properties of certain subclasses of analytic functions defined by a multiplier transformation, Comput. Math. Appl. 5(006), ] M. Fekete and G. Szegö, Eine Bemerkung Uber Ungerade Schlichte Funktionen, J. London Math. Soc. S1-8 no., ] J. L. Liu, Subordination for certain multivalent analytic functions associated with the generalized Srivastava - Attiya operator, Integral Transforms Spec. Funct. 19 (008), ] W. Ma and D. Minda, A Unified treatment of some special classes of univalent functions, in: Proceedings of the Conference on Complex Analysis, Z. Li, F. Ren, L. Yang, and S. Zhang(Eds.), Int. Press (1994), ] G. Murugusundaramoorthy, S. Kavitha and T. Rosy, On the Fekete-Szegö problem for some subclasses of analytic functions defined by convolution, J. Combin. Math. Combin. Comput. 67 (008), 17. )

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