Int. J. Open Problems Complex Analysis, Vol. 3, No. 1, March 2011 ISSN ; Copyright c ICSRS Publication,

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1 Int. J. Open Problems Complex Analysis, Vol. 3, No. 1, March 211 ISSN ; Copyright c ICSRS Publication, Coefficient Estimates for Certain Class of Analytic Functions N. M. Mustafa M. Darus School of Mathematical Sciences, Faculty of Science Technology Universiti Kebangsaan Malaysia Bangi 436 Selangor D. Ehsan, Malaysia nmma 1975@yahoo.com maslina@ukm.my(corresponding author) Abstract In this paper, we consider some coefficient estimates for certain class of analytic functions denoted by CL(β), satisfying K 2 (f, z) (1 β) < 1 β, ( 1 2 < β < 1). Keywords: Analytic functions, convex functions, k-starlike functions, strongly starlike functions, subordination. AMS Mathematics Subject Classification (2): 3C45. 1 Introduction Let H denote the class of analytic functions which are in the open unit disc U = {z : z < 1} on the complex plane C. Let A denote the subclass of H consisting of functions normalized by f() =, f () = 1. Let functions f g be analytic in U. Then a function f is said to be subordinate to a function g, if there exists a function w, analytic in U such that w() =, w(z) < 1 for z < 1 f(z) = g(w(z)). We denote this subordination by (f g). In particular, if a function g is univalent in U we have the following equivalence f(z) g(z) f() = g() f(u) g(u).

2 1 N.M. Mustafa M.Darus Robertson introduced in [1] the classes S (β),c (β) of starlike convex functions of order β 1, in U which is defined by: { } zf S (z) (β) = Re > β (z U), f(z) { } zf C (z) (β) = Re + 1 > β (z U). f(z) If ( β < 1), then a function in either of this set is univalent. If β <, then it may fail to be univalent. In this way, many interesting classes of analytic functions have been studied such as the class SL which is defined by SL = {f A : zf (z) f(z) 1 < 1}, z U. can be found in [5]. And also SL SS ( 1) 2 S, is the class SS (β) of strongly starlike functions of order β given by { SS (β) = f A : Arg zf (z) f(z) < βπ }, ( < β 1), 2 which was studied in [2]. Moreover, k ST SL, for k 2+ 2, where k ST is the class of k-starlike functions introduced in [3] such that { } k ST := f A : Re zf (z) f(z) > k zf (z) f(z) 1, k. Let us denote K(f, z) := zf (z) f (z). In this paper we consider the class CL(β), given by CL(β) := { f A : K 2 (f, z) (1 β) < 1 β }. (1.1) Notice that, L := {w C, R(w) > : w 2 (1 β) < 1 β} is the interior of the right half of the lemniscate of Bernoulli γ : (x 2 +y 2 ) 2 2(1 β)(x 2 y 2 ) =. It can be verified that{w : w 2 1 < 1} L, for( 1 < β < 1), thus 2 SL CL(β). Also, it is easy to see L { } w : Argw < βπ 2, thus CL(β) SS (β).

3 Coefficient estimates for certain class of Corollary 1.1 If k 2 + 2, 1 < β < 1, then 2 k ST CL(β) SS (β) S. Theorem 1.2 The function f belongs to the class CL(β) if only if there exists an analytic function such that q H, q(z) q (z) = (1 β)z + (1 β), q () = 1 β, f(z) = z y q(t) 1 y exp t dtdy. (1.2) Let q 1 (z) = 3 + 2z 3 + z, q 2(z) = 5 + 3z 5 + z, q 3(z) = 8 + 4z 8 + z, q 4 (z) = 8 + 4z 8 + z q 5 (z) = 9 + 5z 9 + z. Now, since q i (z) q (z)for i = 1, 2, 3, 4, we have then by (1.2) that f 1 (z) = z2 2 +z3 9, f 2(z) = 1 2 z z3 + z4 1, f 3(z) = 1 2 z z z z5, 2 Main results f 4 (z) = 1 2 z z z z z6. Theorem 2.1 If a function f(z) = z + a 2 z a n z n belongs to the class CL(β), then k 2 (k 1 2(1 β)) a k 2 2(1 β), ( 1 2 < β < 1). (2.1) Proof. If a function f CL(β), then K(f, z) q (z) = (1 β)z + (1 β). Thus we can write K(f, z) = (1 β)w(z) + (1 β), where w satisfies w() =, w(z) < 1 for z < 1, K(f, z) = zf (z) f (z).

4 12 N.M. Mustafa M.Darus Whence (1 β)(f (z)) 2 = (zf (z)) 2 (1 β)w(z)(f (z)) 2. From this we can obtain 2(1 β)π for < r < 1, = 4(1 β)π 2π k 2 a k 2 r 2k 2 = (1 β) f (re iθ ) 2 dθ 2π (1 β) w(re iθ ) f (re iθ ) 2 dθ 2π (re θ f (re iθ ) 2 (1 β)f (re iθ ) 2 dθ k 2 a k 2 r 2k 2 2π k 2 (k 1) a k 2 r 2k 2, derive into the following inequalities 2(1 β) k 2 a k 2 r 2k 2 k 2 (k 1) a k 2 r 2k 2 2(1 β) k 2 a k 2 r 2k 2 k 2 (k 1) a k 2 r 2k 2. Eventually, if we let r 1, then we obtain k 2 (k 1 2(1 β)) a k 2 2(1 β). Corollary 2.2 If a function f(z) = z +a 2 z 2 +a 3 z a n z n belongs to the class CL(β), then 2(1 β) 1 a k for k 2 k 2 (k 1 2(1 β)) 2 < β < 1. Theorem 2.3 If a function f(z) = a kz k belongs to the class CL(β), then a 2 1, a 3 1, a (2.2) These results are sharp.

5 Coefficient estimates for certain class of Proof: If f(z) = a kz k belongs to the class CL(β), then (1 β)(f (z)) 2 = (zf (z)) 2 (1 β)w(z)(f (z)) 2, where w satisfies w() =, w(z) < 1 for z < 1. Let us denote (zf (z)) 2 = Then we have A k z k, f 2 (z) = B k z k, w(z) = k= C k z k. A k = 4(a 2 ) 2 z a 2 a 3 z 3 + (48a 2 a a 3 2 )z 4 + (8a 2 a a 3 a 2 )z , (2.3) B k = a 1 z 2 + 2a 2 z 3 + (2a 3 + a 2 )z 4 + (2a 2 a 4 a 3 2 a 1 a 5 )z , (2.4) such that A k z k (1 β) B k z k = Thus we obtain k= C k z k B k z k. (2.5) k= A 2 = 4a 2, A 3 = 24a 2 a 3, A 4 = 48a 2 a a 2 3, A 5 = 8a 2 a a 3 a 4, (2.6) B = a 2 1 = 1, B 1 = 4a 1 a 2 = 4a 2, B 2 = 6a 3 a , B 3 = 8a 1 a 4 +12a 2 a 3. (2.7) Equating second third coefficients of (2.5), we obtain (I) (1 β)b 1 = (1 β)b C 1. (II) A 2 (1 β)b 2 = (1 β)(c 2 B + C 1 B 1 ). (III) A 3 (1 β)b 3 = (1 β)(c 3 B + C 2 B 1 + C 1 B 2 ). And so by (2.5), (2.6) (2.7), we have a 2 = 1 4 C 1, a 3 = 1 6 C C2 1, a 4 = 1 8 C 3 + (3 2β 24(1 β) C 1C C2 1 + (2 β 16(1 β) C3 1 Using a well-known inequality C k 1, C k 2 1, we obtain (2.2). Other work related to coefficient problems for different class of analytic functions can be found in ([6],[7],[8]).

6 14 N.M. Mustafa M.Darus 3 Open Problem The method here is employed from the work done by Sokol [4]. It is interesting to see other results in a similar technique for different classes such as the starlike subclasses with respect to symmetric points. We did try, but failed to get any results it is left for the readers to tackle this problem. Acknowledgement: The work here was supported by UKM-ST-6-FRGS References [1] M.S. Robertson, Certain classes of starlike functions, Michigan Math. J, 76 (1) (1954), [2] J. Stankiewicz, Quelques problemes extremaux dans les classes des fonctions a -angulairement etoilees, Ann. Univ. Mariae Curie. Sklodowska, Sect. A, 2 (1966), [3] S. Kanas A. Wisniowska, Conic regions k-uniform convexity, J. Comput. Appl. Math., 15 (1999), [4] J. Sokol, Coefficient estimates in a class of strongly starlike functions, Kyungpook Math. J., 49(29), [5] J. Sokol J. Stankiewicz, Radius of convexity of some subclasses of strongly starlike starlike functions, Folia Scient. Univ. Tech. Res 19(1996), [6] A. Mohammed M. Darus, On Fekete-Szegö problem for certain subclass of analytic functions, Int. J. Open Problems Compt. Math., 3(4) (21), [7] M. H. Al-Abbadi M. Darus, Angular estimates for certain analytic univalent functions, Int. J. Open Problems Complex Analysis, 2(3) (21), [8] M. H. Al-Abbadi M. Darus, The Fekete-Szeg Theorem for a Certain Class of Analytic Functions, Sains Malaysiana, 4(4)(211), in press.