Initial Coefficient Bounds for a General Class of Bi-Univalent Functions

Filomat 29:6 (2015), 1259 1267 DOI 10.2298/FIL1506259O Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://.pmf.ni.ac.rs/filomat Initial Coefficient Bounds for a General Class of Bi-Univalent Functions H. Orhan a, N. Magesh b, V. K. Balaji c a Department of Mathematics, Faculty of Science, Ataturk University, 25240 Erurum, Turkey b Post-Graduate Research Department of Mathematics, Government Arts College for Men, Krishnagiri 635001, Tamilnadu, India c Department of Mathematics, L.N. Govt College, Ponneri, Chennai, Tamilnadu, India Abstract. Recently, Srivastava et al. [22] revieed the study of coefficient problems for bi-univalent functions. Inspired by the pioneering ork of Srivastava et al. [22], there has been triggering interest to study the coefficient problems for the different subclasses of bi-univalent functions (see, for example, [1, 3, 6, 7, 27, 29],). Motivated essentially by the aforementioned orks, in this paper e propose to investigate the coefficient estimates for a general class of analytic bi-univalent functions. Also, e obtain estimates on the coefficients a 2 a 3 for functions in this ne class. Further, e discuss some interesting remarks, corollaries applications of the results presented here. 1. Introduction Let A denote the class of functions of the form = + a n n n=2 hich are analytic in the open unit disk U = { : C < 1}. Further, by S e shall denote the class of all functions in A hich are univalent in U. For analytic functions f in U, f is said to be subordinate to if there exists an analytic function such that (see, for example, [13]) (0) = 0, () < 1 = (()) ( U). This subordination ill be denoted here by f ( U) or, conventionally, by (1) () ( U). 2010 Mathematics Subject Classification. 30C45. Keyords. (Analytic functions, univalent functions, bi-univalent functions, starlike convex functions, bi-starlike bi-convex functions.) Received: 29 November 2013; Accepted: 24 December 2013 Communicated by Hari M. Srivastava Corresponding author: H. Orhan Email addresses: orhanhalit607@gmail.com; horhan@atauni.edu.tr (H. Orhan), nmagi_2000@yahoo.co.in (N. Magesh), balajilsp@yahoo.co.in (V. K. Balaji)

H. Orhan, N. Magesh, V. K. Balaji / Filomat 29:6 (2015), 1259 1267 1260 In particular, hen is univalent in U, f ( U) f (0) = (0) f (U) (U). Some of the important ell-investigated subclasses of the univalent function class S include (for example) the class S (α) of starlike functions of order α (0 α < 1) in U the class K(α) of convex functions of order α (0 α < 1) in U, the class S β P (α) of β spirallike functions of order α (0 α < 1; β < π 2 ), the class S (ϕ) of Ma-Minda starlike functions the class K(ϕ) of Ma-Minda convex functions (ϕ is an analytic function ith positive real part in U, ϕ(0) = 1, ϕ (0) > 0 ϕ maps U onto a region starlike ith respect to 1 symmetric ith respect to the real axis) (see [5, 11, 24]). The above-defined function classes have recently been investigated rather extensively in (for example) [9, 17, 25, 26] the references therein. It is ell knon that every function f S has an inverse f 1, defined by here f 1 ( ) = ( U) f ( f 1 ()) = ( < r 0 ( f ); r 0 ( f ) 1 4 ), f 1 () = a 2 2 + (2a 2 2 a 3) 3 (5a 3 2 5a 2a 3 + a 4 ) 4 +.... A function f A is said to be bi-univalent in U if both f 1 () are univalent in U. Let denote the class of bi-univalent functions in U given by (1). For a brief history interesting examples of functions hich are in (or hich are not in) the class, together ith various other properties of the bi-univalent function class one can refer the ork of Srivastava et al. [22] references therein. In fact, the study of the coefficient problems involving bi-univalent functions as revieed recently by Srivastava et al. [22]. Various subclasses of the bi-univalent function class ere introduced non-sharp estimates on the first to coefficients a 2 a 3 in the Taylor-Maclaurin series expansion (1) ere found in several recent investigations (see, for example, [1 4, 6 8, 12, 14, 16, 19 21, 23, 27, 29]). The aforecited all these papers on the subject ere actually motivated by the pioneering ork of Srivastava et al. [22]. Hoever, the problem to find the coefficient bounds on a n (n = 3, 4,... ) for functions f is still an open problem. Motivated by the aforementioned orks (especially [22] [3, 7]), e define the folloing subclass of the function class. Definition 1.1. Let h : U C, be a convex univalent function such that h(0) = 1 h( ) = h() ( U R(h()) > 0). Suppose also that the function h() is given by h() = 1 + B n n n=1 ( U). A function given by (1) is said to be in the class NP µ,λ (β, h) if the folloing conditions are satisfied: ( ) µ ( ) µ 1 f, e iβ (1 λ) + λ f () h() cos β + i sin β ( U), (2) ( ) µ ( ) µ 1 () () e iβ (1 λ) + λ () h() cos β + i sin β ( U), (3)

H. Orhan, N. Magesh, V. K. Balaji / Filomat 29:6 (2015), 1259 1267 1261 here β ( π/2, π/2), λ 1, µ 0 the function is given by () = a 2 2 + (2a 2 2 a 3) 3 (5a 3 2 5a 2a 3 + a 4 ) 4 +... (4) the extension of f 1 to U. Remark 1.2. If e set h() = 1+A 1+B, 1 B < A 1, in the class NPµ,λ(β, h), e have NPµ,λ as ( ) µ ( ) µ 1 f, e iβ (1 λ) + λ f () 1 + A cos β + i sin β ( U) 1 + B ( ) µ ( ) µ 1 () () e iβ (1 λ) + λ () 1 + A cos β + i sin β ( U), 1 + B here β ( π/2, π/2), λ 1, µ 0 the function is given by (4). 1+A (β, 1+B ) defined Remark 1.3. Taking h() = 1+(1 2α), 0 α < 1 in the class NP µ,λ (β, h), e have NPµ,λ(β, α) f NPµ,λ(β, α) if the folloing conditions are satisfied: ( ) µ ( ) µ 1 f, R eiβ (1 λ) + λ f () > α cos β ( U) ( ) µ ( ) µ 1 () () R eiβ (1 λ) + λ () > α cos β ( U), here β ( π/2, π/2), 0 α < 1, λ 1, µ 0 the function is given by (4). It is interest to note that the class µ,λ (0, α) := N (α) the class as introduced studied by Çağlar et al. [3]. NP µ,λ Remark 1.4. Taking λ = 1 h() = 1+(1 2α), 0 α < 1 in the class NP µ,λ (β, h), e have NPµ,1(β, α) f NP µ,1 (β, α) if the folloing conditions are satisfied: ( ) µ 1 f, R eiβ f () > α cos β ( U) ( ) µ 1 () R eiβ () > α cos β ( U), here β ( π/2, π/2), 0 α < 1, µ 0 the function is given by (4).We notice that the class NP µ,1 (0, α) := F (µ, α) as introduced by Prema Keerthi [16]. Remark 1.5. Taking µ + 1 = λ = 1 h() = 1+(1 2α), 0 α < 1 in the class NP µ,λ (β, h), e have NP0,1(β, α) f NP 0,1 (β, α) if the folloing conditions are satisfied: ( f, R e iβ f ) () > α cos β ( U)

H. Orhan, N. Magesh, V. K. Balaji / Filomat 29:6 (2015), 1259 1267 1262 ( ) R e iβ () > α cos β ( U), () here β ( π/2, π/2), 0 α < 1 the function is given by (4). In addition, the class NP 0,1 (0, α) := S (α) as studied by Li Wang [10] considered by others. Remark 1.6. Taking µ = 1 h() = 1+(1 2α), 0 α < 1 in the class NP µ,λ (β, h), e have NP1,λ(β, α) f NP 1,λ (β, α) if the folloing conditions are satisfied: ( f, R (e iβ (1 λ) )) + λ f () > α cos β ( U) ( R (e iβ (1 λ) () )) + λ () > α cos β ( U), here β ( π/2, π/2), 0 α < 1, λ 1 the function is given by (4). Further, the class NP 1,λ (0, α) := B (α, λ) as introduced discussed by Frasin Aouf [6] Remark 1.7. Taking µ = λ = 1 h() = 1+(1 2α), 0 α < 1 in the class NP µ,λ (β, h), e have NP1,1(β, α) f NP 1,1 (β, α) if the folloing conditions are satisfied: f, R ( e iβ f () ) > α cos β ( U) R ( e iβ () ) > α cos β ( U), here β ( π/2, π/2), 0 α < 1 the function is given by (4). Also, the class NP 1,1 (0, α) := H α as introduced studied by Srivastava et al. [22]. In order to derive our main result, e have to recall here the folloing lemmas. Lemma 1.8. [15] If p P, then p i 2 for each i, here P is the family of all functions p, analytic in U, for hich here R{p()} > 0 ( U), p() = 1 + p 1 + p 2 2 + ( U). Lemma 1.9. [18, 28] Let the function ϕ() given by ϕ() = B n n ( U) n=1 be convex in U. Suppose also that the function h() given by ψ() = ψ n n ( U) n=1 is holomorphic in U. If then ψ() ϕ() ( U) ψ n B 1 (n N = {1, 2, 3,... }).

H. Orhan, N. Magesh, V. K. Balaji / Filomat 29:6 (2015), 1259 1267 1263 The object of the present paper is to introduce a general ne subclass NP µ,λ (β, h) of the function class obtain estimates of the coefficients a 2 a 3 for functions in this ne class NP µ,λ (β, h). 2. Coefficient Bounds for the Function Class NP µ,λ (β, h) In this section e find the estimates on the coefficients a 2 a 3 for functions in the class NP µ,λ (β, h). Theorem 2.1. Let given by (1) be in the class NP µ,λ (β, h), λ 1 µ 0, then 2 B 1 cos β (1 + µ)(2λ + µ) (5) 2 B 1 cosβ (2λ + µ)(1 + µ), (6) Proof. It follos from (2) (3) that there exists p, q P such that ( ) µ ( ) µ 1 e iβ (1 λ) + λ f () = p() cos β + i sin β (7) ( ) µ ( ) µ 1 () () e iβ (1 λ) + λ () = p() cos β + i sin β, (8) here p() h() q() h() have the forms p() = 1 + p 1 + p 2 2 +... ( U) (9) q() = 1 + q 1 + q 2 2 +... ( U). (10) Equating coefficients in (7) (8), e get e iβ (λ + µ)a 2 = p 1 cos β [ a 2 ] e iβ 2 2 (µ 1) + a 3 (2λ + µ) = p 2 cos β (12) e iβ (λ + µ)a 2 = q 1 cos β [ ] e iβ (µ + 3) a2 2 2 a 3 (2λ + µ) = q 2 cos β. (14) (11) (13) From (11) (13), e get p 1 = q 1 (15)

H. Orhan, N. Magesh, V. K. Balaji / Filomat 29:6 (2015), 1259 1267 1264 2e i2β (λ + µ) 2 a 2 2 = (p2 1 + q2 1 ) cos2 β. (16) Also, from (12) (14), e obtain a 2 2 = e iβ (p 2 + q 2 ) cos β (1 + µ)(2λ + µ). (17) Since p, q h(u), applying Lemma 1.9, e immediately have p m = p (m) (0) m! B 1 (m N), (18) q m = q (m) (0) m! B 1 (m N). (19) Applying (18), (19) Lemma 1.9 for the coefficients p 1, p 2, q 1 q 2, e readily get 2 B 1 cos β (1 + µ)(2λ + µ). This gives the bound on a 2 as asserted in (5). Next, in order to find the bound on a 3, by subtracting (14) from (12), e get 2(a 3 a 2 2 )(2λ + µ) = e iβ (p 2 q 2 ) cos β. (20) It follos from (17) (20) that a 3 = e iβ cos β(p 2 + q 2 ) (1 + µ)(2λ + µ) + e iβ (p 2 q 2 ) cos β. (21) 2(2λ + µ) Applying (18), (19) Lemma 1.9 once again for the coefficients p 1, p 2, q 1 q 2, e readily get 2 B 1 cosβ (2λ + µ)(1 + µ). This completes the proof of Theorem 2.1. 3. Corollaries Consequences In vie of Remark 1.2, if e set h() = 1 + A 1 + B h() = 1 + (1 2α) 1 ( 1 B < A 1; U) (0 α < 1; U), in Theorem 2.1, e can readily deduce Corollaries 3.1 3.2, respectively, hich e merely state here ithout proof.

H. Orhan, N. Magesh, V. K. Balaji / Filomat 29:6 (2015), 1259 1267 1265 Corollary 3.1. Let given by (1) be in the class NP µ,λ 2(A B) cos β (1 + µ)(2λ + µ) 1+A (β, 1+B ), then (22) 2(A B)cosβ (2λ + µ)(1 + µ), (23) here β ( π/2, π/2), µ 0 λ 1. Corollary 3.2. Let given by (1) be in the class NP µ,λ (β, α), 0 α < 1, µ 0 λ 1, then 4(1 α) cos β (1 + µ)(2λ + µ) (24) 4(1 α)cosβ (2λ + µ)(1 + µ), (25) Remark 3.3. When β = 0 the estimates of the coefficients a 2 a 3 of the Corollary 3.2 are improvement of the estimates obtained in [3, Theorem 3.1]. Corollary 3.4. Let given by (1) be in the class NP µ,1 (β, α), 0 α < 1 µ 0, then 4(1 α) cos β (1 + µ)(2 + µ) (26) 4(1 α)cosβ (2 + µ)(1 + µ), (27) Corollary 3.5. Let given by (1) be in the class NP 0,1 (β, α), 0 α < 1, then 2(1 α) cos β (28) 2(1 α)cosβ, (29) Remark 3.6. Taking β = 0 in Corollary 3.5, the estimate (28) reduces to a 2 of [10, Corollary 3.3] (29) is improvement of a 3 obtained in [10, Corollary 3.3].

H. Orhan, N. Magesh, V. K. Balaji / Filomat 29:6 (2015), 1259 1267 1266 Corollary 3.7. Let given by (1) be in the class NP 1,λ (β, α), 0 α < 1 λ 1, then 2(1 α) cos β 2λ + 1 (30) 2(1 α)cosβ, 2λ + 1 (31) Remark 3.8. Taking β = 0 in Corollary 3.7, the inequality (31) improves the estimate of a 3 in [6, Theorem 3.2]. Corollary 3.9. Let given by (1) be in the class NP 1,1 (β, α), 0 α < 1, then 2(1 α) cos β 3 (32) 2(1 α)cosβ, 3 (33) Remark 3.10. For β = 0 the inequality (33) improves the estimate a 3 of [22, Theorem 2]. References [1] R. M. Ali, S. K. Lee, V. Ravichran, S. Supramanian, Coefficient estimates for bi-univalent Ma-Minda starlike convex functions, Appl. Math. Lett. 25 (2012), no. 3, 344 351. [2] S. Bulut, Coefficient estimates for a class of analytic bi-univalent functions, Novi Sad J. Math. 43 (2013), no. 2, 59 65. [3] M. Çağlar, H. Orhan N. Yağmur, Coefficient bounds for ne subclasses of bi-univalent functions, Filomat, 27 (2013), no. 7, 1165 1171. [4] E. Deni, Certain subclasses of bi-univalent functions satisfying subordinate conditions, J. Classical Anal. 2 (1) (2013), 49 60. [5] P. L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer, Ne York, 1983. [6] B. A. Frasin M. K. Aouf, Ne subclasses of bi-univalent functions, Appl. Math. Lett. 24 (2011), no. 9, 1569 1573. [7] S.P.Goyal P.Gosami, Estimate for initial Maclaurin coefficients of bi-univalent functions for a class defined by fractional derivatives, J. Egyptian Math. Soc., 20 (2012), 179 182. [8] T. Hayami S. Oa, Coefficient bounds for bi-univalent functions, Pan Amer. Math. J. 22 (4) (2012), 15 26. [9] Y. C. Kim H. M. Srivastava, Some subordination properties for spirallike functions, Appl. Math. Comput. 203 (2008), no. 2, 838 842. [10] X.-F. Li A.-P. Wang, To ne subclasses of bi-univalent functions, Internat. Math. Forum 7 (2012), 1495 1504. [11] W. C. Ma, D. Minda, A unified treatment of some special classes of univalent functions, in: Proceedings of the Conference on Complex Analysis, Tianjin, 1992, 157 169, Conf. Proc. Lecture Notes Anal. I, Int. Press, Cambridge, MA, 1994. [12] N. Magesh J. Yamini, Coefficient bounds for certain subclasses of bi-univalent functions, Internat. Math. Forum 8 (2013), no. 27, 1337 1344. [13] S. S. Miller P. T. Mocanu, Differential subordinations, Monographs Textbooks in Pure Applied Mathematics, 225, Dekker, Ne York, 2000. [14] G. Murugusundaramoorthy, N. Magesh V. Prameela, Coefficient bounds for certain subclasses of bi-univalent functions, Abs. Appl. Anal., Volume 2013, Article ID 573017, 3 pages. [15] Ch. Pommerenke, Univalent Functions, Venhoeck & Ruprecht, Göttingen, 1975. [16] S. Prema, B. S. Keerthi, Coefficient bounds for certain subclasses of analytic functions, J. Math. Anal. 4 (2013), no. 1, 22 27. [17] V. Ravichran, Y. Polatoglu, M. Bolcal A. Sen, Certain subclasses of starlike convex functions of complex order, Hacettepe J. Math. Stat. 34 (2005), 9 15. [18] W. Rogosinski, On the coefficients of subordinate functions, Proc. London Math. Soc. (2) 48 (1943), 48 82. [19] S. Sivaprasad Kumar, V. Kumar V. Ravichran, Estimates for the initial coefficients of bi-univalent functions, arxiv:1203.5480v1, 2012.

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