ON THE FEKETE-SZEGÖ INEQUALITY FOR A CLASS OF ANALYTIC FUNCTIONS DEFINED BY USING GENERALIZED DIFFERENTIAL OPERATOR

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1 Acta Universitatis Apulensis ISSN: No. 6/0 pp 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 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

2 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

3 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

4 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 ) 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

5 ) 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

6 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

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

8 = 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

9 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

10 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

11 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-FRGS References F. M. Al-Oboudi, On univalent functions defined by a generalized Sǎlǎgean operator, Int. J. Math. Math. Sci., 7 004), M. Darus and R. W. Ibrahim, On subclasses for generalized operators of complex order, Far East Journal of Math. Sci.FJMS), 333) 009), B. Frasin, and M. Darus, On Fekete- Szegö problem using Hadamard products, Intern. Math. Jour., 3 003), 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), S. Owa and H. M. Srivastava, Univalent and starlike generalized hypergeometric functions, Canad J. Math., 39 5) 987),

12 6 H. M. Srivastava and A. K. Mishra, Applications of fractional calculus to parabolic starlike and uniformly convex functions, Comput. Math. Appl., ), 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), 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 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, Salma Faraj Ramadan School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia salma.naji@gmail.com Maslina Darus School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia maslina@ukm.my 78