Entisar El-Yagubi & Maslina Darus

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1 Journal of Quality Measurement Analysis Jurnal Pengukuran Kualiti dan Analisis JQMA 0() ON FEKETE-SZEGÖ PROBLEMS FOR A SUBCLASS OF ANALYTIC FUNCTIONS (Berkenaan Permasalahan Fekete-Szegö bagi Subkelas Fungsi Analisis) ENTISAR EL-YAGUBI & MASLINA DARUS ABSTRACT The aim of this paper is to determine the Fekete-Szegö inequalities for a normalised analytic function defined on the open unit disc for which z(d λ ) / (D λ )δ N 0 λ λ 0 lies in a region starlike with respect to it is symmetric with respect to the real axis by using the operator D λ given recently by the authors. As a special case of this result Fekete-Szegö inequality for a class of functions defined by fractional derivatives is also obtained. Keywords: analytic function; starlike function; subordination; Fekete-Szegö inequality; derivative operator ABSTRAK Matlamat makalah ini adalah untuk menentukan ketaksamaan Fekete-Szegö bagi fungsi analisis ternormal yang ditakrif pada cakera unit terbuka dengan z(d λ ) / (D λ ) λ λ 0 δ N 0 terletak pada rantau bak bintang terhadap dan simetri terhadap paksi nyata dengan menggunakan pengoperasi D λ yang diberi penulis baru-baru ini. Sebagai kes khas bagi hasil ini ketaksamaan Fekete-Szegö bagi kelas fungsi yang ditakrif oleh terbitan pecahan juga diperoleh. Kata kunci: fungsi analisis; fungsi bak bintang; subordinasi; ketaksamaan Fekete-Szegö; pengoperasi terbitan. Introduction Let A denote the class of functions of the form = z a n z n (z U ) () n= which are analytic in the open unit disc U = {z : z C z < }. Also let S be the subclass of A consisting of all functions which are univalent in U. Let φ P where φ(z) is an analytic function with positive real part on A with φ(0) = φ (0) > 0 let S * (φ) be the class of functions in f A such that z f (z) φ(z) (z U ) () C(φ) be the class of functions in f A for which

2 Entisar El-Yagubi & Maslina Darus z f (z) f (z) φ(z) (z U ). (3) where denotes to the subordination between two analytic functions. Let a n be a complex number 0 µ. A classical theorem of Fekete Szegö (933) states that for f S given by () µa exp µ µ. The inequality is sharp. For a brief history of the Fekete-Szegö problem for the class of starlike functions S convex functions C close-to-convex functions K see the papers by Mohammed Darus (00) Srivastava et al. (00) Darus (00) Al-Abbadi Darus (0) Ravichran et al. (004) Al-Shaqsi Darus (008). In particular for f K given by () Keogh Merkes (969) showed that 3 4µ if 0 µ 3 µa 3 4 if 9µ 3 µ 3 if 3 µ 4µ 3 if µ for each µ there is a function in K for which equality holds. Definition. (El-Yagubi & Darus 03) Let f be in the class A. For δ N 0 λ λ 0 we define the differential operator as follows: D λ = z (λ λ )(n ) b n= (n ) b m C(δ n)a n z n (4) where C(δ n) = defined by δ n δ = (δ ) n (n )! (δ ) n denotes the Pochhammer symbol n = 0 (δ ) n = δ (δ )(δ )...(δ n ) δ!n!. 8

3 On Fekete-Szegö problems for a subclass of analytic functions Using the operator D λ we define the class M λ (φ) as follows: Definition.. Let φ P be a univalent starlike function with respect to which maps the unit disc U onto a region in the right half plane symmetric with respect to the real axis φ(0) = φ (0) > 0. A function f A is in the class M λ (φ) if z(d λ ) D λ φ(z) (5) where δ N 0 λ λ 0 D λ denotes the differential operator (4). The motivation of this paper is to generalise the Fekete-Szegö inequalities proved by Srivastava Mishra (000) for functions in the class M λ (φ). We also give some applications of our results for certain functions defined by fractional derivatives. To prove our main results the following lemma is required. Lemma. (Ma & Minda 994). If p (z) = c (z) c z is an analytic function with positive real part in U then 4v if v 0 c vc if 0 v 4v if v. When v < 0 or v > the equality holds if only if p (z) is z z rotations. If 0 < v < then equality holds if only if p (z) is z If v = 0 the equality holds if only if p (z) = γ z z γ z (0 γ ) z or one of its or one of its rotations. z or one of its rotations. If v = the equality holds if only if p (z) is the reciprocal of one of the functions such that the equality holds in the case of v = 0. Also the above upper bound is sharp it can be improved as follows when 0 < v < : c vc v c (0 < v ) c vc ( v) c ( < v ). 9

4 Entisar El-Yagubi & Maslina Darus. Main Results Our first result is contained in the following theorem. Theorem.. Let φ(z) be an analytic function with positive real part on A φ(z) = B z B z. If is given by () belongs to M λ φ(z) then ( λ b) m B (δ )(δ )( (λ ) b) ( λ b) m µb m (δ ) ( λ b) m ( λ b) m B if µ σ (δ )(δ )( (λ ) b) m ( λ µa b) m B if σ (δ )(δ )( (λ ) b) m µ σ ( λ b) m B (δ )(δ )( (λ ) b) ( λ b) m µb m (δ ) ( λ b) m ( λ b) m B if µ σ (δ )(δ )( (λ ) b) m (6) where σ σ (δ ) ( λ b) m (B B ) B (δ )(δ )( λ b) m ( (λ ) b) m B (δ ) ( λ b) m (B B ) B (δ )(δ )( λ b) m ( (λ ) b) m B. (8) (7) Proof. For f M λ φ(z) let p(z) = z(d λ ) D λ = b z b z. (9) Substituting (4) in (9) comparing the coefficients of z z 3 on both sides in equation (9) we have λ b b m (λ ) b λ b (δ )a = b m (δ )(δ )a 3 = λ λ b b m (δ ) a b. (0) 0

5 On Fekete-Szegö problems for a subclass of analytic functions Now we want to find out the values for b b. Since φ(z) is univalent p φ the function p (z) = φ ( p(z)) φ ( p(z)) = c z c z () is analytic has a positive real part in U. Thus we have p(z) = φ p (z) p (z). From the equations (9) () we obtain c b z b z = φ z c z c z c z this implies = φ c z (c c )z = B c z B (c c )z B 4 c z () b = B c b = B (c c ) 4 B c. By substituting the values of b b in equation (0) we have B a = c ( b) m ( λ b) m (δ ) a 3 = Therefore we have ( 4 B c B (c c ) 4 B c )( λ b) m ( (λ ) b) m (δ )(δ ) B a 3 µa = ( λ b) m ( (λ ) b) m (δ )(δ ) (c c [ ( B B. (3) (δ )(δ )( λ b) m ( (λ ) b) m µ ( λ b) m (δ ) ( λ b) m (δ ) B )])

6 Entisar El-Yagubi & Maslina Darus which implies B a 3 µa = ( λ b) m ( (λ ) b) m (δ )(δ ) [c νc ] where ν = ( B B (δ )(δ )( λ b)m ( (λ ) b) m µ ( λ b) m (δ ) ( λ b) m (δ ) B ). If µ σ then by Lemma. we obtain ( λ µa b) m B (δ )(δ )( (λ ) b) ( λ b) m µb m (δ ) ( λ b) m ( λ b) m B (δ )(δ )( (λ ) b) m. If µ σ then we get ( λ µa b) m B (δ )(δ )( (λ ) b) ( λ b) m µb m (δ ) ( λ b) m ( λ b) m B (δ )(δ )( (λ ) b) m. Similarly if σ µ σ we get µa ( λ b) m B (δ )(δ )( (λ ) b) m.

7 On Fekete-Szegö problems for a subclass of analytic functions 3. Application of Fractional Derivatives For fixed g A let M g λ (φ) be the class of functions f A for which ( f g) M λ (φ). Note that for any two analytic functions = z a n z n g(z) = z b n z n their convolution is defined by n= ( f * g)(z) = * g(z) = z a n b n z n. n= Definition 3. (Owa & Srivastava 987). Let f be analytic in a simply connected region of the z-plane containing the origin. The functional derivative of f of order γ is defined by D z γ = Γ( γ ) d z dz 0 f (ζ ) (z ζ ) γ dζ (0 γ < ) where the multiplicity of (z ζ ) γ is removed by requiring that log(z ζ ) is real for z ζ > 0. Using Definition 3. the well known extension involving fractional derivatives fractional integrals Owa Srivastava (987) introduced the operator Ω γ : A A which is defined by Ω γ = Γ( γ )z γ D z γ (γ 34 ). n= γ The class M λ γ that M λ Let (φ) consists of functions f A for which Ω γ f M λ (φ). Note (φ) is the special case of the class M g λ (φ) when g(z) = z n= Γ(n )Γ( γ ) Γ(n γ ) g(z) = z g n z n (g n > 0). n= z n. Since D λ M λ g (φ) if only if D λ * g(z) M λ (φ) we obtain the coefficient estimate for functions in the class M g λ (φ) from the corresponding estimate for functions in the class M λ (φ). Applying Theorem. for the function 3

8 Entisar El-Yagubi & Maslina Darus D λ * g(z) = z λ λ b b m (δ )a g z we get the following result after an obvious change of the parameter µ. Theorem 3.. Let g(z) = z g n= n z n (g n > 0) let the function φ(z) be given by φ(z) = B n z n. If D λ given by (4) belongs to M g λ (φ) then n= µa ( λ b) m B (δ )(δ )( (λ ) b) m g 3 ( λ b) m µb (δ ) ( λ b) m g ( λ b) m B if µ σ (δ )(δ )( (λ ) b) m g 3 ( λ b) m B (δ )(δ )( (λ ) b) m g 3 if σ µ σ ( λ b) m B ( λ b) m µb (δ )(δ )( (λ ) b) m g 3 (δ ) ( λ b) m g ( λ b) m B if µ σ (δ )(δ )( (λ ) b) m g 3 where σ g (δ ) ( λ b) m (B B ) B g 3 (δ )(δ )( λ b) m ( (λ ) b) m B Since σ (Ω γ D λ g (δ ) ( λ b) m (B B ) B g 3 (δ )(δ )( λ b) m ( (λ ) b) m B. ) = z n= Γ(n )Γ( γ ) Γ(n γ ) (λ )(n ) b (n ) b m C(δ n)a n z n we have g Γ(3)Γ( γ ) Γ(3 γ ) = ( γ ) 4

9 On Fekete-Szegö problems for a subclass of analytic functions g 3 Γ(4)Γ( γ ) Γ(4 γ ) = 6 ( γ )(3 γ ). Proof. By using the same technique as in the proof of Theorem. the required result is obtained. For g g 3 given by above equalities Theorem 3. reduces to the following result. Corollary 3.. Let g(z) = z g n= n z n (g n > 0) let the function φ(z) be given by φ(z) = B n z n. If D λ given by (3) belongs to M g λ (φ). Then n= ( γ )(3 γ )( λ b) m B 6(δ )(δ )( (λ ) b) ( γ ) ( λ b) m µb m 4(δ ) ( λ b) m ( γ )(3 γ )( λ b) m B 6(δ )(δ )( (λ ) b) if µ σ m ( γ )(3 γ )( λ µa b) m B 6(δ )(δ )( (λ ) b) if σ µ σ m ( γ )(3 γ )( λ b) m B 6(δ )(δ )( (λ ) b) ( γ ) ( λ b) m µb m 4(δ ) ( λ b) m ( γ )(3 γ )( λ b) m B 6(δ )(δ )( (λ ) b) if µ σ m where (3 γ )(δ ) ( λ σ b) m (B B ) B 3( γ )(δ )(δ )( λ b) m ( (λ ) b) m B σ (3 γ )(δ ) ( λ b) m (B B ) B 3( γ )(δ )(δ )( λ b) m ( (λ ) b) m B. Acknowledgement The work presented here was partially supported by AP

10 Entisar El-Yagubi & Maslina Darus References Al-Abbadi M. H. & Darus M. 0. The Fekete-Szegö theorem for a certain class of analytic functions. Sains Malaysiana 40(4): Al-Shaqsi K. & Darus M On Fekete-Szegö problems for certain subclass of analytic functions. Appl. Math. Sci. (9): Darus M. 00. The Fekete-Szegö problems for close-to-convex functions of the class K sh (αβ). Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis 8(): 3-8. El-Yagubi E. & Darus M. 03. Subclasses of analytic functions defined by new generalised derivative operator. Journal of Quality Measurement Analysis 9(): Fekete M. & Szegö G Eine bermerkung über ungerade schlichte function. J. Lond. Math. Soc. 8: Keogh F. R. & Merkes E. P A coefficient inequality for certain classes of analytic functions. Proc. Amer. Math. Soc. 0: 8-. Ma W. & Minda D A unified treatment of some special classes of univalent functions. In Li Z. Ren F. Yang L. & Zhang S. (Eds.). Proceedings of the Conference on Complex Analysis pp Int. Press. Mohammed A. & Darus M. 00. On Fekete-Szegö problem for certain subclass of analytic functions. Int. J. Open Problems Compt. Math. 3(4): Owa S. & Srivastava H. M Univalent starlike generalized hypergeometric functions. Canad. J. Math. 39(5): Ravichran V. Gangadharan A. & Darus M On Fekete-Szegö inequality for certain class of Bazilevic functions. Far East Journal of Mathematical Sciences 5(): Srivastava H. M. & Mishra A. K Applications of fractional calculus to parabolic starlike uniformly convex functions. Computer Math. Appl. 39: Srivastava H. M. Mishra A. K. & Das M. K. 00. The Fekete-Szegö problem for a subclass of close-to-convex functions. Complex Variables Theory Application 44: School of Mathematical Sciences Faculty of Science Technology University Kebangsaan Malaysia UKM Bangi Selangor DE MALAYSIA entisar_e980@yahoo.com maslina@ukm.edu.my* *Corresponding author 6