COEFFICIENT BOUNDS FOR A SUBCLASS OF BI-UNIVALENT FUNCTIONS

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1 TWMS J. Pure Appl. Math., V.6, N.2, 205, pp COEFFICIENT BOUNDS FOR A SUBCLASS OF BI-UNIVALENT FUNCTIONS ŞAHSENE ALTINKAYA, SIBEL YALÇIN Abstract. An analytic function f defined on the open unit disk U = { : < } is biunivalent if the function f and its inverse f are univalent in U. Inspired by the recent work of Hamidi et al. [8], we propose to investigate the coefficient estimates for a general class of analytic and bi-univalent functions. Also, we obtain estimates on the coefficients a 2, a 3 and a n for functions in this class. Some earlier results are shown to be special cases of our results. Keywords: analytic and univalent functions, bi-univalent functions, Salagean derivative. AMS Subject Classification: 30C45, 30C50.. Introduction Let A denotes the class of functions f which are analytic in the open unit disk U = { : < } with in the form f() = + a n n. () Let S be the subclass of A consisting of the form () which are also univalent in U and let P be the class of functions p() = + p n n that are analytic in U and satisfy the condition Re (p()) > 0 in U. By the Caratheodory s lemma (e.g., see [7] ) we have p n 2. Let f A. We define the differential operator D k, k N 0 = N {0}, where N = {, 2,...}, by (see [2]) D 0 f () = f () ; D f () = Df () = f ();. ( ) D k f () = D D k f(). For k N 0, 0 β <, λ 0, we introduce the subclass Q (k, λ, β) of S of functions of the form () satisfying the condition { ( λ) D k f () + λd k+ } f () Re > β, U, (2) where D k stands for Salagean derivative introduced by Salagean [2].For f A, the class Q (k, λ, β) S and was first defined and investigated by Porwal and Darus []. Department of Mathematics, Faculty of Arts and Science, Uludag University, Bursa, Turkey sahsenealtinkaya@gmail.com Manuscript received April

2 Ş. ALTINKAYA, S. YALÇIN: COEFFICIENT BOUNDS FOR A SUBCLASS... 8 It is easy to see that Q (k, { λ, β) Q (k, λ 2, β) for λ > λ 2 0. } Thus, for λ, 0 β <, Q (k, λ, β) Q (k,, β) = f A : Re Dk+ f () > β; 0 β <. It is well known that every f S has an inverse f, defined by f (f ()) =, ( U) and where f ( f (w) ) = w, ( w < r 0 (f), r 0 (f) ), 4 f (w) = w a 2 w 2 + ( 2a 2 2 a 3 ) w 3 ( 5a 3 2 5a 2 a 3 + a 4 ) w 4 +. A function f () A is said to be bi-univalent in U if both f () and f () are univalent in U. Finding bounds for the coefficients of classes of bi-univalent functions dates back to 967 (see [0]). But the interest on the bounds for the coefficients of classes of bi-univalent functions picked up by the publications of Brannan and Taha [6], and Srivastava et al. [3]. Not much is known about behavior of higher order coefficients of classes of bi-univalent functions, as Ali et al. [5]. Motivating with their work, we let f Q (k, λ, β) and g = f Q (k, λ, β) and use the Faber polynomial coefficient expansion to provide bounds for the general coefficients a n of such functions with a given gap series [8]. 2. Main results Using the Faber polynomial expansion of functions f A of the form (), the coefficients of its inverse map g = f may be expressed as, [3], where g (w) = f (w) = w + n K n n (a 2, a 3,...) w n, (3) K n n = ( 2n + )! (n )! an 2 + [2 ( n + )]! (n 3)! an 3 2 a ( 2n + 3)! (n 4)! an 4 2 a 4 + [ + [2 ( n + 2)]! (n 5)! an 5 2 a5 + ( n + 2) a 2 ] 3 + (4) + ( 2n + 5)! (n 6)! an 6 2 [a 6 + ( 2n + 5) a 3 a 4 ] + + a n j 2 V j, j 7 such that V j with 7 j n is a homogeneous polynomial in the variables a 2, a 3,..., a n [4]. In particular, the first three terms of K n n are

3 82 TWMS J. PURE APPL. MATH., V.6, N.2, K 2 3 K 3 4 K 4 = a 2, 2 = 2a 2 2 a 3, (5) 3 = ( 5a 3 2 5a 2 a 3 + a 4 ). In general, for any p N, an expansion of K p n is as, [3], Kn p p (p ) = pa n + En where En p = En p (a 2, a 3,...) and by [], E m n (a, a 2,..., a n ) = p! (p 3)!3! E3 n p! (p n)!n! En n, (6) m! (a ) µ... (a n ) µn, (7) µ!...µ n! while a =, and the sum is taken over all nonnegative integers µ,..., µ n satisfying m= Evidently, E n n (a, a 2,..., a n ) = a n, [2]. µ + µ µ n = m, (8) µ + 2µ nµ n = n. Theorem 2.. For 0 β <, λ and k N 0, let f Q (k, λ, β) and g Q (k, λ, β). If a m = 0 ; 2 m n, then a n 2 ( β) n k [ + (n ) λ] ; n 4 (9) Proof. For analytic functions f of the form () we have ( λ) D k f () + λd k+ f () = + n k [ + (n ) λ] a n n, (0) and for its inverse map, g = f, we have ( λ) D k g (w) + λd k+ g (w) w = + = + n k [ + (n ) λ] b n w n = n k [ + (n ) λ] () n K n n (a 2, a 3,..., a n ) w n. On the other hand, since f Q (k, λ, β) and g = f Q (k, λ, β) by definition, there exist two positive real part functions p () = + c n n and q (w) = + d n w n where Rep () > 0 and Req (w) > 0 in P so that ( λ) D k f () + λd k+ f () ( λ) D k g (w) + λd k+ g (w) w = + ( β) Kn (c, c 2,..., c n ) n, (2) = + ( β) Kn (d, d 2,..., d n ) w n. (3)

4 Ş. ALTINKAYA, S. YALÇIN: COEFFICIENT BOUNDS FOR A SUBCLASS Comparing the corresponding coefficients of (0) and (2) yields n k [ + (n ) λ] a n = ( β) K n (c, c 2,..., c n ), (4) and similarly, from () and (3) we obtain n nk [ + (n ) λ] K n n (b 0, b,..., b n ) = ( β) K n (d, d 2,..., d n ). (5) Note that for a m = 0 ; 2 m n we have b n = a n and so n k [ + (n ) λ] a n = ( β) c n (6) n k [ + (n ) λ] a n = ( β) d n Now taking the absolute values of either of the above two equations and applying the Caratheodory s lemma, we obtain a n ( β) c n n k [ + (n ) λ] = ( β) d n n k [ + (n ) λ] 2 ( β) n k [ + (n ) λ]. (7) Theorem 2.2. For 0 β <, λ + 2, let f Q (k, λ, β) and g Q (k, λ, β). Then 2 ( β) (i) a 2 3 k ( + 2λ), 0 β < 3k (+2λ) 2 2k 2 (+2λ) ; 3 k (+2λ) 2 ( β) 3 2 k, k (+2λ) 2 2k (+2λ) 2 β <. ( + 2λ) 3 k (+2λ) 2 ( β) (ii) a 3 3 k ( + 2λ). (8) (iii) a3 2a 2 2 ( β) 2 3 k ( + 2λ). Proof. Replacing n by 2 and 3 in (4) and (5), respectively, we find that 2 k ( + λ) a 2 = ( β) c, (9) 3 k ( + 2λ) a 3 = ( β) c 2, (20) 2 k ( + λ) a 2 = ( β) d, (2) Dividing (9) or (2), by 2 k ( + λ), taking their absolute values, and applying the Caratheodory s lemma, we obtain a 2 ( β) c 2 k ( + λ) = ( β) d 2 ( β) 2 k ( + λ) 2 k ( + λ). (22) Adding (20) to (22) implies or, equivalently, 2.3 k ( + 2λ) a 2 2 = ( β) (c 2 + d 2 ) (23) a 2 2 = ( β) (c 2 + d 2 ) 2.3 k. (24) ( + 2λ) An application of Caratheodory s lemma followed by taking the square roots yields a 3 = ( β) c 2 2 ( β) 3 k ( + 2λ) 3 k ( + 2λ). (25)

5 84 TWMS J. PURE APPL. MATH., V.6, N.2, 205 In the following, dividing (22) by ( + 2λ), taking the absolute values of both sides, and applying the Caratheodory s lemma, we obtain a3 2a 2 ( β) d 2 ( β) 2 = 3 k ( + 2λ) 3 k ( + 2λ). (26) Remark 2.. If we put k = 0 in Theorem, we obtain the corresponding results due to Jahangiri and Hamidi [9]. References [] Airault, H., (2007), Symmetric sums associated to the factoriation of Grunsky coefficients, in Conference, Groups and Symmetries, Montreal, Canada, April. [2] Airault, H., (2008), Remarks on Faber polynomials, Int. Math. Forum, 3(9), pp [3] Airault, H., Bouali, A., (2006), Differential calculus on the Faber polynomials, Bulletin des Sciences Mathematiques, 30(3), pp [4] Airault, H., Ren, J., (2002), An algebra of differential operators and generating functions on the set of univalent functions, Bulletin des Sciences Mathematiques, 26(5), pp [5] Ali, R.M., Lee, S.K., (202), Ravichandran V., and Supramaniam S., Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions, Applied Mathematics Letters, 25(3), pp [6] Brannan, D.A., Taha, T.S., (986), On some classes of bi-univalent functions, Studia Universitatis Babeş- Bolyai. Mathematica, 3(2), pp [7] Duren, P.L., (983), Univalent Functions, Grundlehren Math. Wiss. Springer, New York, 259. [8] Hamidi, G.S., Halim, S.A., Jahangiri, J.M., (203), Faber polynomial coefficient estimates for meromorphic bi-starlike functions, International Journal of Mathematics and Mathematical Sciences, Article ID 49859, 4p. [9] Jahangiri, J.M., Hamidi, G.S., (203), Coefficient estimates for certain classes of bi-univalent functions, International Journal of Mathematics and Mathematical Sciences, Article ID 90560, 4p. [0] Lewin, M., (967), On a coefficient problem for bi-univalent functions, Proc. Amer. Math. Soc., 8, pp [] Porwal, S., Darus, M., (203), On a new subclass of bi-univalent functions, J. Egypt. Math. Soc., 2(3), pp [2] Salagean, G.S., (983), Subclasses of univalent functions, in: Complex Analysis - Fifth Romanian Finish Seminar, Bucharest,, pp [3] Srivastava, H.M., Mishra, A.K., Gochhayat, P., (200), Certain subclasses of analytic and bi-univalent functions, Applied Mathematics Letters, 23(0), pp Şahsene Altınkaya was born in 990. She received her B.S. degree in mathematics (202) from Erciyes University and her M.S. degree in mathematics (204) from Uludag University, Turkey. She is a Research Assistant of the Department of Mathematics Faculty of Arts and Science, Uludag University since 203. Her research areas include geometric function theory, analytic functions and bi-univalent functions.

6 Ş. ALTINKAYA, S. YALÇIN: COEFFICIENT BOUNDS FOR A SUBCLASS Sibel Yalçin received her Ph.D. degree in Mathematics in 200 from the Uludag University of Bursa, Turkey. She became a full Professor in 20. She is currently with the Department of Mathematics, Uludag University. Her research interests include harmonic mappings, geometric function theory, meromorphic functions, analytic functions, bi- univalent functions, convolution operators.