Correspondence should be addressed to Serap Bulut;

Transcription

1 The Scientific World Journal Volume 201, Article ID 17109, 6 pages Research Article Coefficient Estimates for Initial Taylor-Maclaurin Coefficients for a Subclass of Analytic and Bi-Univalent Functions Defined by Al-Oboudi Differential Operator Serap Bulut Civil Aviation College, Kocaeli University, Arslanbey Campus, İmit-Kocaeli, Turkey Correspondence should be addressed to Serap Bulut; bulutserap@yahoo.com Received 5 August 201; Accepted 7 October 201 Academic Editors: H. Bulut and J. Park Copyright 201 Serap Bulut. This is an open access article distributed under the Creative Commons Attribution License, hich permits unrestricted use, distribution, and reproduction in any medium, provided the original ork is properly cited. We introduce and investigate an interesting subclass (n,β;h)of analytic and bi-univalent functions in the open unit disk U. For functions belonging to the class (n, β; h), e obtain estimates on the first to Taylor-Maclaurin coefficients a 2 and a. 1. Introduction Let R = (, ) be the set of real numbers, C the set of complex numbers, and N := {1, 2,,... = N 0 \ {0 (1) the set of positive integers. Let A denotetheclassofallfunctionsoftheform f () =+ k=2 hich are analytic in the open unit disk a k k (2) U = { : C and <1. () We also denote by S the class of all functions in the normalied analytic function class A hich are univalent in U. For to functions f and g,analyticinu, e say that the function f is subordinate to g in U and rite f () g() ( U), (4) if there exists a Schar function ω, hich is analytic in U ith ω (0) =0, ω () <1, ( U) (5) such that Indeed, it is knon that f () =g(ω ()), ( U). (6) f () g(), ( U) f (0) =g(0), f (U) g(u). Furthermore, if the function g is univalent in U,thenehave the folloing equivalence: f () g(), ( U) f (0) =g(0), f (U) g(u). For f A, Al-Oboudi [1]introducedthefolloingoperator: (7) (8) D 0 δf () =f(), (9) D 1 δ f () = (1 δ) f () +δf () =: D δ f (), (δ 0), (10) D n δ f () =D δ (D n 1 δ f ()), (n N). (11)

2 2 The Scientific World Journal If f is given by (2), then from (10)and(11)eseethat D n δ f () =+ [1+(k 1)δ] n a k k, (n N 0 ), (12) k=2 ith D n δf(0) = 0. Whenδ=1,egetSǎlǎgean s differential operator D n 1 =Dn,[2]. Since univalent functions are one-to-one, they are invertible and the inverse functions need not be defined on the entire unit disk U. In fact, the Koebe one-quarter theorem [] ensures that the image of U under every univalent function f S contains a disk of radius 1/4. Thuseveryfunction f Ahas an inverse f 1, hich is defined by f 1 (f ()) = ( U), f(f 1 ()) = ( <r 0 (f) ; r 0 (f) 1 (1) 4 ). In fact, the inverse function f 1 is given by f 1 () = a 2 2 +(2a 2 2 a ) (14) (5a 2 5a 2a +a 4 ) 4 +. Afunctionf Ais said to be bi-univalent in U if both f and f 1 are univalent in U. Let denote the class of biunivalent functions in U given by (2). For a brief history and interesting examples of functions in the class, see[4] (see also [5, 6]). In fact, the aforecited ork of Srivastava et al. [4] essentially revived the investigation of various subclasses of the bi-univalent function class in recent years; it as folloed by such orks as those by Frasin and Aouf [7], Poral and Darus [8], and others (see, e.g., [9 17]). Motivated by the abovementioned orks, e define the folloing subclass of function class. Definition 1. Let h:u C be a convex univalent function such that h (0) =1, h() = h () ( U; R (h ()) >0). (15) Afunctionf, defined by (2), is said to be in the class (n, β; h) if the folloing conditions are satisfied: f, e iβ ((1 λ) Dn δf () f ()) ) h() cos β+isin β ( U), e iβ ((1 λ) Dn δg () g ()) ) h() cos β+isin β ( U), (16) here β ( π/2, π/2), λ 1,thefunctiong is given by g () = a (2a 2 2 a ) (5a 2 5a 2a +a 4 ) 4 +, and D n δ is the Al-Oboudi differential operator. (17) Remark 2. If e set h () = 1+A 1+B ( 1B<A1) (18) class denoted by (n,β;a,b) hich is the subclass of the functions f satisfying e iβ ((1 λ) Dn δf () ) 1+A cos β+isin β ( U), 1+B e iβ ((1 λ) Dn δg () g ()) ) 1+A cos β+isin β ( U), 1+B (19) here β ( π/2, π/2), λ 1, thefunctiong is defined by (17), and D n δ is the Al-Oboudi differential operator. Remark. If e set h () = (0 α<1) (20) class denoted by (n, β, α) hich is the subclass of the functions f satisfying R {e iβ ((1 λ) Dn δf () f ()) ) > α cos β ( U), R {e iβ ((1 λ) Dn δg () g ()) ) > α cos β ( U), (21) here β ( π/2, π/2), λ 1, thefunctiong is defined by (17), and D n δ is the Al-Oboudi differential operator. Remark 4. If e set δ=1, h() = (0 α<1) (22) class denoted by NP λ (n, β, α) hich is the subclass of the functions f satisfying R {e iβ ((1 λ) Dn f () +λ(d n f ()) ) > α cos β ( U), R {e iβ ((1 λ) Dn g () +λ(d n g ()) ) > α cos β ( U), (2) here β ( π/2, π/2), λ 1, thefunctiong is defined by (17), and D n is the Sǎlǎgean differential operator.

3 The Scientific World Journal Remark 5. If e set n=0, h() = (0 α<1) (24) class denoted by NP λ (β, α) hich is the subclass of the functions f satisfying R {e iβ ((1 λ) R {e iβ ((1 λ) f () +λf ()) > α cos β ( U), g () +λg ()) > α cos β ( U), (25) here β ( π/2, π/2), λ 1,andthefunctiong is defined by (17). Remark 6. If e set n = 0, λ = 1, h () = (0 α<1) (26) class denoted by NP (β, α) hich is the subclass of the functions f satisfying R {e iβ f () >αcos β ( U), R {e iβ g () >αcos β ( U), (27) here β ( π/2, π/2) and the function g is defined by (17). We note that NP λ (n, 0, α) = H (n, α, λ) (see [8]), NP λ (0, α) = B (α, λ) (see [7]), NP (0, α) = H (α) (see [4]). (28) Firstly, in order to derive our main results, e need the folloing lemma. Lemma 7 (see [18]). Let the function h() given by h () = n=1 B n n (29) be convex in U. Suppose also that the function φ() given by φ () = n=1 c n n (0) is holomorphic in U.Ifφ() h() ( U),then c n B 1 (n N). (1) The object of the present paper is to find estimates on the Taylor-Maclaurin coefficients a 2 and a for functions in this ne subclass (n,β;h)of the function class. 2. A Set of General Coefficient Estimates In this section, e state and prove our general results involving the bi-univalent function class (n,β;h) given by Definition 1. Theorem 8. Let the function f() given by the Taylor-Maclaurin series expansion (2) be in the function class (n,β;h) (β ( π/2, π/2),λ 1,δ 0,n N 0) (2) ith h () =1+B 1 +B () a 2 min { B 1 cos β {(1+δ) n (1+λ), B 1 cos β (1+2δ) n (1+2λ), { (4) B 1 cos β (1+2δ) n (1+2λ). (5) Proof. It follos from (16)that e iβ ((1 λ) Dn δf () f ()) ) (6) =p() cos β+isin β ( U), e iβ ((1 λ) Dn δg () g ()) ) =q() cos β+isin β ( U), (7) here p() h() and q() h() have the folloing Taylor-Maclaurin series expansions: p () =1+p 1 +p 2 2 +, (8) q () =1+q 1 +q 2 2 +, (9) respectively. No, upon equating the coefficients in (6)and (7), e get e iβ (1+δ) n (1+λ) a 2 =p 1 cos β, (40) e iβ (1+2δ) n (1+2λ) a =p 2 cos β, (41) e iβ (1+δ) n (1+λ) a 2 =q 1 cos β, (42) e iβ [ (1+2δ) n (1+2λ) a +2(1+2δ) n (1+2λ) a 2 2 ] =q 2 cos β. From (40)and(42), e obtain (4) p 1 = q 1, (44) 2e 2iβ (1+δ) 2n (1+λ) 2 a 2 2 =(p2 1 +q2 1 ) cos2 β. (45)

4 4 The Scientific World Journal Also, from (41)and(4), e find that a 2 2 = e iβ (p 2 +q 2 ) cos β 2(1+2δ) n (1+2λ). (46) Since p, q h(u), according to Lemma 7, eimmediately have p k = q k = p (k) (0) k! B 1 (k N), q (k) (0) k! B 1 (k N). (47) Applying (47)andLemma 7 for the coefficients p 1, p 2, q 1,and q 2,fromtheequalities(45)and(46), e obtain a 2 2 a 2 2 B 1 2 cos 2 β (1+δ) 2n (1+λ) 2, (48) B 1 cos β (1+2δ) n (1+2λ), (49) respectively. So e get the desired estimate on the coefficient a 2 as asserted in (4). Next, in order to find the bound on the coefficient a,e subtract (4)from(41). We thus get 2(1+2δ) n (1+2λ) a 2(1+2δ) n (1+2λ) a 2 2 =e iβ (p 2 q 2 ) cos β. (50) Upon substituting the value of a 2 2 from (45) into(50), it follos that a = e 2iβ (p 2 1 +q2 1 ) cos2 β 2(1+δ) 2n (1+λ) 2 So e get + e iβ (p 2 q 2 ) cos β 2(1+2δ) n (1+2λ). (51) B 1 2 cos 2 β (1+δ) 2n (1+λ) 2 + B 1 cos β (1+2δ) n (1+2λ). (52) On the other hand, upon substituting the value of a 2 2 from (46)into(50), it follos that a = e iβ (p 2 +q 2 ) cos β 2(1+2δ) n (1+2λ) + e iβ (p 2 q 2 ) cos β 2(1+2δ) n (1+2λ). (5) And e get B 1 cos β (1+2δ) n (1+2λ). (54) Comparing the inequalities in (52) and(54) completes the proof of Theorem 8.. Corollaries and Consequences h () = 1+A ( 1B<A1) (55) 1+B in Theorem 8,ehavethefolloingcorollary. Corollary 9. Let the function f() given by the Taylor- (n,β;a,b) (β ( π/2, π/2),λ 1,δ 0, 1B<A1,n N 0 ). (56) a (A B) cos β 2 min { (1+δ) n (1+λ), (A B) cos β (1+2δ) n (1+2λ), (A B) cos β (1+2δ) n (1+2λ). (57) h () = (0 α<1) (58) in Theorem 8,ehavethefolloingcorollary. Corollary 10. Let the function f() given by the Taylor- (n,β,α) (β ( π/2, π/2),λ 1,δ 0,0α<1,n N 0 ). a 2 min { (1+δ) n (1+λ), (1+2δ) n (1+2λ), (1+2δ) n (1+2λ). (59) (60) δ=1, h() = (0 α<1) (61) in Theorem 8,ehavethefolloingcorollary. Corollary 11. Let the function f() given by the Taylor- NP λ (n, β, α) (β ( π/2, π/2),λ 1,0α<1,n N 0 ). (62)

5 The Scientific World Journal 5 a 2 (1 α) cos β min {2 2 n, (1+λ) n, (1+2λ) n. (1+2λ) (6) Remark 12. When β=0, Corollary 11 is an improvement of the folloing estimates obtained by Poral and Darus [8]. Corollary 1 (see [8]). Let the function f() given by the Taylor- H (n, α, λ) (λ 1, 0α<1, n N 0 ). (64) a 2 2 (1 α) n (1+2λ), 4(1 α)2 2 2n (1+λ) (1 α) n (1+2λ). (65) n=0, h() = (0 α<1) (66) in Theorem 8,ehavethefolloingcorollary. Corollary 14. Let the function f() given by the Taylor- NP λ (β, α) (β ( π/2, π/2),λ 1,0α<1). (67) a 2 min { { {, 1+λ. 1+2λ 1+2λ, (68) Remark 15. When β=0, Corollary 14 is an improvement of thefolloingestimatesobtainedbyfrasinandaouf[7]. Corollary 16 (see [7]). Let the function f() given by the Taylor- B (α, λ) (λ 1,0α<1). (69) a 2 2 (1 α) 1+2λ, 4(1 α)2 (1+λ) (1 α) 1+2λ. (70) n = 0, λ = 1, h () = in Theorem 8,e have the folloing corollary. (0 α<1) (71) Corollary 17. Let the function f() given by the Taylor- NP (β, α) (β ( π/2, π/2),0α<1). (72) a 2 min { {(1 α) cos β, {., (7) Remark 18. When β=0, Corollary 17 is an improvement of the folloing estimates obtained by Srivastava et al. [4]. Corollary 19 (see [4]). Let the function f() given by the Taylor- References H (α) (0α<1). (74) a 2 2 (1 α), (1 α)(5 α). (75) [1] F. M. Al-Oboudi, On univalent functions defined by a generalied Sălăgean operator, International Mathematics and Mathematical Sciences, vol.2004,no.27,pp , [2] G.S.Sǎlǎgean, Subclasses of univalent functions, in Complex Analysis-Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981), vol. 101ofLecture Notes in Mathematics, pp , Springer, Berlin, Germany, 198. []P.L.Duren,Univalent Functions, vol.259ofgrundlehren der Mathematischen Wissenschaften, Springer, Ne York, NY, USA, 198. [4] H. M. Srivastava, A. K. Mishra, and P. Gochhayat, Certain subclasses of analytic and bi-univalent functions, Applied Mathematics Letters, vol. 2, no. 10, pp , [5] D.A.BrannanandT.S.Taha, Onsomeclassesofbi-univalent functions, in Mathematical Analysis and Its Applications, S. M.Mahar,A.Hamoui,andN.S.Faour,Eds.,vol.ofKFAS Proceedings Series, pp. 5 60, Pergamon Press, Elsevier Science, Oxford, UK, [6] D.A.BrannanandT.S.Taha, Onsomeclassesofbi-univalent functions, Studia Universitatis Babeş-Bolyai Mathematica, vol. 1, no. 2, pp , 1986.

6 6 The Scientific World Journal [7] B. A. Frasin and M. K. Aouf, Ne subclasses of bi-univalent functions, Applied Mathematics Letters,vol.24,no.9,pp , [8] S. Poral and M. Darus, On a ne subclass of bi-univalent functions, the Egyptian Mathematical Society,vol.21, no., pp , 201. [9] S. Bulut, Coefficient estimates for a class of analytic and biunivalent functions, Novi Sad Mathematics, vol. 4, no. 2, pp , 201. [10] S. Bulut, Coefficient estimates for ne subclasses of analytic and bi-univalent functions defined by Al-Oboudi differential operator, Function Spaces and Applications, vol. 201, Article ID 18192, 7 pages, 201. [11] M. Çağlar, H. Orhan, and N. Yağmur, Coefficient bounds for ne subclasses of bi-univalent functions, Filomat,vol.27,no.7, pp , 201. [12] S. P. Goyal and P. Gosami, Estimate for initial Maclaurin coefficients of bi-univalent functions for a class defined by fractional derivatives, the Egyptian Mathematical Society, vol. 20,no.,pp ,2012. [1] T. Hayami and S. Oa, Coefficient bounds for bi-univalent functions, Pan-American Mathematical Journal, vol.22,no.4, pp , [14] H. Orhan, N. Magesh, and V. K. Balaji, Initial coefficient bounds for a general class of bi-univalent functions, Preprint. [15] H. M. Srivastava, S. Bulut, M. Çağlar, and N. Yağmur, Coefficient estimates for a general subclass of analytic and biunivalent functions, Filomat,vol.27,no.5,pp ,201. [16] Q. H. Xu, Y. C. Gui, and H. M. Srivastava, Coefficient estimates for a certain subclass of analytic and bi-univalent functions, Applied Mathematics Letters,vol.25,no.6,pp ,2012. [17] Q.H.Xu,H.G.Xiao,andH.M.Srivastava, Acertaingeneral subclass of analytic and bi-univalent functions and associated coefficient estimate problems, Applied Mathematics and Computation,vol.218,no.2,pp ,2012. [18] W. Rogosinski, On the coefficients of subordinate functions, Proceedings of the London Mathematical Society, vol.48,no.2, pp.48 82,194.

7 Advances in Operations Research Advances in Decision Sciences Applied Mathematics Algebra Probability and Statistics The Scientific World Journal International Differential Equations Submit your manuscripts at International Advances in Combinatorics Mathematical Physics Complex Analysis International Mathematics and Mathematical Sciences Mathematical Problems in Engineering Mathematics Discrete Mathematics Discrete Dynamics in Nature and Society Function Spaces Abstract and Applied Analysis International Stochastic Analysis Optimiation