Acta Universitatis Apulensis ISSN: 58-539 No. 6/0 pp. 67-78 ON THE FEKETE-SZEGÖ INEQUALITY FOR A CLASS OF ANALYTIC FUNCTIONS DEFINED BY USING GENERALIZED DIFFERENTIAL OPERATOR Salma Faraj Ramadan, Maslina Darus Abstract. In this present investigation, the Fekete- Szegö inequality for certain normalized analytic functions f defined on the open unit disk for which zdk α,β,λ,δ fz)) Dα,β,λ,δ k fz) k N 0, α, β, λ, δ 0) lies in a region starlike with respect to and is symmetric with respect to the real axis will be obtained. In addition, certain applications of the main result for a class of functions defined by convolution are given. As a special case of this result, Fekete- Szegö inequality for a class of functions defined by fractional derivative is obtained. The motivation of this paper is due to the work given by Srivastava and Mishra in 6. 000 Mathematics Subject Classification: 30C45. Key words: Analytic function, Starlike functions, Derivative operator, Fekete- Szegö inequality.. Introduction Let A be the class of functions f of the form f z) = z + a n z n ) which are analytic in the open unit disk U = {z : z C and z < }. Further, let S denote the class of functions which are univalent in U. For a function f A, we define D 0 f z) = f z) 67
D α,β,λ,δ f z) = λ δ) β α) f z) + λ δ) β α) zf z) = z + λ δ) β α) n ) + a n z n D k α,β,λ,δ f z) = D α,β,λ,δ. D k α,β,λ,δ f z) ) Dα,β,λ,δ k f z) = z + λ δ) β α) n ) + k a n z n ) for α 0, β 0, λ 0, δ 0, λ > δ, β > α) and k {0,,,...}. Remark. i) When α = 0, δ = 0, λ =, β =, it reduces to Sǎlǎgean differential operator9. ii) When α = 0, reduces to Darus and Ibrahim differential operator. iii) And when α = 0, δ = 0, β =, reduces to Al- Oboudi differential operator. Let φ z) be an analytic function with positive real part on U with φ z) =, φ z) > 0 which maps the unit disk U onto a region starlike with respect to which is symmetric with respect to the real axis. Let S φ) be the class of functions in f z) S for which zf z) φ z), z U) f z) and C φ) be the class of functions in f z) S for which + zf z) f z) φ z), z U) where denotes the subordination between analytic functions. These classes were investigated and studied by Ma and Minda 4. They have obtain the Fekete- Szegö inequality for the functions in the class C φ). Since f C φ) if and only if zf z) S φ), we get the Fekete- Szegö inequality for functions in the class S φ). For a brief history of the Fekete- Szegö problem for class of starlike, convex, and close to convex functions, see the recent paper by Srivastava et.al 7. In the present paper, we obtain the Fekete- Szegö inequality for the class Mα,β,λ,δ k φ) as defined below. Also we give applications of our result to certain functions defined through convolution or Hadamard product) and in particular we consider a class Mα,β,λ,δ k φ) 68
defined by fractional derivatives. The object of this paper is to generalize the Fekete- Szegö inequality of that given by Srivastava and Mishra 6. Definition. Let φ z) be a univalent starlike function with respect to which maps the unit disk U onto a region in the right half plane which is symmetric with respect to the real axis, φ 0) = and φ 0) > 0. A function f A is in the class Mα,β,λ,δ k φ) if ) z Dα,β,λ,δ k f z) Dα,β,λ,δ k f z) φ z). For fixed g A, we define the class M k,g α,β,λ,δ φ) to be the class of functions f A for which f g) Mα,β,λ,δ k φ). In order to derive our main results, we have to recall here the following lemma 4: Lemma. If p z) = + c z + c z +... is an analytic function with positive real part in U, then c νc 4ν + if ν 0; if 0 ν ; 4ν if ν. When ν < 0 or ν >, the equality holds if and only if p z) is + z)/ z) or one of its rotations. If 0 < ν <, then the equality holds if and only if p z) is + z )/ z ) or one of its rotations. If ν = 0, the equality holds if and only if ) + γ + z p z) = z + γ ) z, 0 γ ) + z or one of its rotations. If ν =, the equality holds if and only if p z) is the reciprocal of one of the functions such that the equality holds in the case of ν = 0. Also the above upper bound is sharp, and it can be improved as follows when 0 < ν < : c νc + ν c 0 < ν ) and c νc + ν) c ) < ν..fekete-szegö problem Our main result is the following: Theorem. Let φ z) = + B z + B z +... If f z) given by ) belongs to 69
M k α,β,λ,δ φ), then a3 µa B λ δ)β α)+ k µb λ δ)β α)+ k + B λ δ)β α)+ k if µ σ ; B λ δ)β α)+ k ifσ µ σ ; 3) B + λ δ)β α)+ k µb λ δ)β α)+ k B λ δ)β α)+ k if µ σ, where The result is sharp. Proof. For f Mα,β,λ,δ k φ), let σ := λ δ) β α) + { k B B ) + B } λ δ) β α) + k B σ := λ δ) β α) + { k B + B ) + B } λ δ) β α) + k. B p z) = ) z Dα,β,λ,δ k f z) D k α,β,λ,δ f z) = + b z + b z +.... 4) From 4), we obtain λ δ) β α) + k a = b, λ δ) β α) + k a 3 = λ δ) β α) + k a + b. 5) Since φ z) is univalent and p φ, the function is analytic and positive real in U. Also we have p z) = + φ p z)) φ p z)) = + c z + c z +... p z) = φ ) p z), 6) p z) + 70
) and from this equality and 4), + b z + b z +... = φ c z+c z +... +c z+c z +... = φ c z + c ) c z +... ) = +B c z+b c ) c z +...+B... we obtain b = B c and b = B c ) c + Where ν = a 3 µa = B 4 λ δ) β α) + k c c 4 c z + 4 B c. Therefore, we have { B B )} λ δ) β α) + k µ λ δ) β α) + k λ δ) β α) + k B, B B a 3 µa = B { c 4 λ δ) β α) + k νc }. ) λ δ) β α) + k µ λ δ) β α) + k λ δ) β α) + k B. If µ σ, then by applying Lemma, we get a 3 µa B λ δ) β α) + k µb λ δ) β α) + k + B λ δ) β α) + k which is the first part of assertion 3). Next, if µ σ, by applying Lemma, we get a 3 µa B λ δ) β α) + k + µb λ δ) β α) + k B λ δ) β α) + k. if µ = σ, then equality holds if and only if ) ) + γ + z γ z p z) = z +, 0 γ, z U) + z or one of its rotations. if µ = σ, then B B ) λ δ) β α) + k µ λ δ) β α) + k λ δ) β α) + k B = 0. Therefore, p z) = + γ ) + z z + γ 7 ) z, 0 < γ <, z U). + z
Finally, we see that and a 3 µa = B 4 λ δ) β α) + k c c { B B } λ δ) β α) + k µ λ δ) β α) + k λ δ) β α) + k B max B B ) λ δ) β α) + k µ λ δ) β α) + k λ δ) β α) + k B, Therefore using Lemma, we get σ µ σ ). a3 µa B c = 4 λ δ) β α) + k B λ δ) β α) + k, σ µ σ ). If σ < µ < σ, then we have z p z) = p z) = + νz, 0 ν ). νz Our result now follows by an application of Lemma. To show that the bounds are sharp, we define the function Ks φ s =, 3,...) by ) Dα,β,λ,δ k Kφ s z) D k α,β,λ,δ Kφ s z) = φ z s ), K φ s 0) = 0 = K φ s 0) and the function F γ and G γ, 0 γ ) by ) z Dα,β,λ,δ k F γ z) ) z z + ν) Dα,β,λ,δ k F = φ, F γ 0) = 0 = F γ 0) γ z) + νz and ) z Dα,β,λ,δ k G γ z) D k α,β,λ,δ G γ z) ) z z + ν) = φ + νz, G γ 0) = 0 = G γ 0). 7
Clearly the functions K φ s, F γ, G γ M k α,β,λ,δ φ). Also we write Kφ = K φ. If µ < σ or µ > σ, then the equality holds if and only if f is Ks φ or one of its rotations. When σ < µ < σ, the equality holds if and only if f is K φ 3 or one of its rotations. If µ = σ, then the equality holds if and only if f is F γ or one of its rotations. If µ = σ, then the equality holds if and only if f is G γ or one of its rotations. Remark. If σ µ σ, then in view of Lemma, Theorem can be improved. Let σ 3 be given by If σ µ σ 3, then If σ 3 µ σ, then σ 3 := λ δ) β α) + k B + B ) λ δ) β α) + k. B a3 µa λ δ) β α) + k + λ δ) β α) + k B µ λ δ) β α) + k λ δ) β α) + k B B + λ δ) β α) + k B B λ δ) β α) + k. a 3 µa λ δ) β α) + k + λ δ) β α) + k B B + B µ λ δ) β α) + k λ δ) β α) + k λ δ) β α) + k B B λ δ) β α) + k. Proof. For the values of σ µ σ 3, we have a 3 µa + µ σ ) a = a a B 4 λ δ) β α) + k c νc B + µ σ ) 4 λ δ) β α) + k c 73
= B 4 λ δ) β α) + k c νc + + µ λ δ) β α) { +k B B ) + B }) B λ δ) β α) + k B 4 λ δ) β α) + k c = = { B λ δ) β α) + k c νc + ν c } Similarly, for the value of σ 3 µ σ, we write a 3 µa + σ µ) a = B λ δ) β α) + k. + B 4 λ δ) β α) + k c νc B + σ µ) 4 λ δ) β α) + k c B = 4 λ δ) β α) + k c νc + λ δ) β α) + k { B + B ) + B } λ δ) β α) + k B µ ) B 4 λ δ) β α) + k c = = { B c λ δ) β α) + k νc + ν) c } Thus, Remark holds. B λ δ) β α) + k. 3. Applications to Functions Defined by Fractional Derivatives For two analytic functions f z) = z + a n z n and g z) = z + b n z n, their convolution or Hadamard product) is defined to be the function f g) z) given by f g) z) = f z) g z) = z + a n b n z n. 74
Definition.8 Let f be analytic in a simply-connected region of the z-plane containing the region. The fractional derivative of f of order γ is defined by Dz γ d f z) Γ γ) dz z 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 the above definition and its known extensions involving fractional derivatives and fractional integrals, Owa and Srivastava 5 introduced the operator Ω γ : A A defined by Ω γ f) z) = Γ γ) z γ D γ z f z), γ, 3, 4,...). The class M k,γ α,β,λ,δ φ) consists of functions f A for which Ωγ f Mα,β,λ,δ k φ). Note that M k,γ k,g α,β,λ,δ φ) is the special case of the class Mα,β,λ,δ φ) when Let Since g z) = z + g z) = z + Γ n + ) Γ γ) z n. 7) Γ n + γ) g n z n, g n > 0). Dα,β,λ,δ k f z) = z + λ δ) β α) n ) + k a n z n M g α,β,λ,δ φ) if and only if D k α,β,λ,δ f g ) z) = z + λ δ) β α) n ) + k a n g n z n Mα,β,λ,δ k φ) 8) we obtain the coefficient estimate for functions in the class M k,g α,β,λ,δ φ), from the corresponding estimate for functions in the class Mα,β,λ,δ k φ). Applying Theorem for the operator 8), we get the following Theorem after an obvious change of the parameter µ : Theorem. Let g z) = z + g k z k, g k > 0) and let the function φ z) be given by φ z) = + n= k= B n z n. If operator ) belongs to M k,g α,β,λ,δ φ), then 75
a 3 µa B g 3 λ δ)β α)+ k B g 3 λ δ)β α)+ k g 3 B + λ δ)β α)+ k µg 3 B λ δ)β α)+ k + µg 3 B λ δ)β α)+ k B λ δ)β α)+ k B λ δ)β α)+ k if µ σ ; ifσ µ σ ; if µ σ, where σ := g λ δ) β α) + { k B B ) + B } g 3 λ δ) β α) + k B σ := g λ δ) β α) + { k B + B ) + B } g 3 λ δ) β α) + k. B The result is sharp. Since Ω γ D k α,β,λ,δ f ) z) = z + we have and g 3 : g := Γ n + ) Γ γ) Γ n + γ) Γ 3) Γ γ) Γ 3 γ) Γ 4) Γ 3 γ) Γ 4 γ) = λ δ) β α) n ) + k a n z n = γ 9) 6 γ) 3 γ). 0) For g and g 3 given by 9) and 0), Theorem reduces to the following: Theorem 3. Let g z) = z + g k z k, by φ z) = + a 3 µa n= k= g k > 0) and let the function φ z) be given B n z n. If Dα,β,λ,δ k k,γ f given by ) belongs to Mα,β,λ,δ φ), then 76
γ)3 γ) 6 γ)3 γ) 6 γ)3 γ) 6 B λ δ)β α)+ k B λ δ)β α)+ k B + λ δ)β α)+ k 3 γ)µb 3 γ)λ δ)β α)+ k + 3 γ)µb 3 γ)λ δ)β α)+ k B if µ σ λ δ)β α)+ k ; B λ δ)β α)+ k ifσ µ σ ; if µ σ, where σ := 3 γ) λ δ) β α) + { k B B ) + B } 3 γ) λ δ) β α) + k B σ := 3 γ) λ δ) β α) + { k B + B ) + B } 3 γ) λ δ) β α) + k. B The result is sharp. Remark 3. When k = 0, λ =, β =, α = 0, δ = 0, B = 8 and B π = 6 the above Theorem 3 reduces to a recent result of Srivastava and Mishra 6, 3π Theorem 8, P.64) for a class of functions for which Ω γ f z) is a parabolic starlike functions see 3,5). Note also, other work related to the upper bounds of the Fekete-Szegó theorem can be found in 0. Acknowledgement: The work here is fully supported by UKM-ST-06-FRGS044-00. References F. M. Al-Oboudi, On univalent functions defined by a generalized Sǎlǎgean operator, Int. J. Math. Math. Sci., 7 004), 49-436. M. Darus and R. W. Ibrahim, On subclasses for generalized operators of complex order, Far East Journal of Math. Sci.FJMS), 333) 009), 99-308. 3 B. Frasin, and M. Darus, On Fekete- Szegö problem using Hadamard products, Intern. Math. Jour., 3 003), 89-95. 4 W. Ma and D. Minda, A unified treatment of some speial classes of univalent functions, in: Proceeding of the conference on complex analysis, Z. Li. F. Ren, L. Yang, and S. Zhang Eds.), Int. Press 994), 57-69. 5 S. Owa and H. M. Srivastava, Univalent and starlike generalized hypergeometric functions, Canad J. Math., 39 5) 987), 057-077. 77
6 H. M. Srivastava and A. K. Mishra, Applications of fractional calculus to parabolic starlike and uniformly convex functions, Comput. Math. Appl., 39 000), 57-69. 7 H. M. Srivastava, A. K. Mishra and M. K. Das, The Fekete- Szegö problem for a subclass of close-to-convex functions, Complex Variables Theory Appl., 44 00), 45-63. 8 H. M. Srivastava and S. Owa, Univalent functions, fractional calculus and their applications., John Wiley and Sons, New Jersey, 989). 9 G. S. Sǎlǎgean, Subclasses of univalent functions, Lecture Notes in Math.03, Springer, Verlag Berlin, 983), PP.36-37. 0 S. P. Goyal and Pranay Goswami, Certain coefficient inequalities for Sakaguchi type functions and applications to fractional derivative operator, Acta Universitatis Apulensis, No. 9/009, 59-66. Salma Faraj Ramadan School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia E-mail: salma.naji@gmail.com Maslina Darus School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia E-mail: maslina@ukm.my 78