Coefficient Estimate for Two Subclasses with Certain Close-to-Convex Functions

Transcription

1 Int. Journal of Math. Analysis, Vol. 5, 2011, no. 8, Coefficient Estimate for Two Subclasses with Certain Close-to-Convex Functions Liangpeng Xiong The College of Engineering and Technical of Chengdou University of Technology, Leshan, Sichuan, , China Xiaoli Liu The College of Engineering and Technical of Chengdou University of Technology, Leshan, Sichuan, , China Abstract Let J(A, B) denotes the class of functions f which are analytic in the unit disc E = z : z < 1} and Satisfying zf (z) h(z) 1+Az 1+Bz, 1 A<B 1,z E, where h(z) is convex univalent in E.In this paper, we consider two classes L(A, B, C, D) and M(A, B, C, D), which are consisting of analytic functions and satisfying (zf (z)) 1+Cz 2(zf (z)) (G(z) G( z)) 1+Cz 1+Dz H(z) 1+Dz and, respectively, where H(z),G(z) J(A, B), 1 D<C 1.The aim of paper is to determine coefficient estimate for the classes L(A, B, C, D) and M(A, B, C, D). Mathematics Subject Classification: 30C45 Keywords: Analytic functions, Close-to-convex function, Subordination, Coefficient estimates 1 Introduction Let U be the class of functions which are analytic in the open unit disk E = z : z < 1} given by ω(z) = r k z k (1)

2 382 Liangpeng Xiong and Xiaoli Liu and satisfying the conditions ω(0) = 0 and ω(z) < 1,z E. Let S denotes the class of functions f(z) which are analytic and univalent in E of the form f(z) =z + a k z k,z E (2) k=2 A function f(z) S is said to be in the class J(A, B) if it exists a convex univalent function h(z) such that zf (z) h(z) 1+Az, 1 A<B 1,z E (3) 1+Bz The class J(A, B) is introduced and discussed by B.S. Mehrok and Gagandeep Singh [3]. In fact, J(1, 1) J was investigated by Gawad and Thomas [1] earlier. Obviously J(A, B) is a subclass of the class J of close-to-convex function. Now we need to definite the following two classes of functions with J(A, B): Definition 1.1 A function f(z) S is said to be in the class L(A, B, C, D) if there exists a function H(z) such that (zf (z)) H(z) 1+Cz, 1 D<C 1,z E (4) 1+Dz where H(z) =z + b k z k J(A, B), 1 B<A 1 (5) k=2 Definition 1.2 A function f(z) S is said to be in the class M(A, B, C, D) if there exists a function G(z) such that 2(zf (z)) 1+Cz, 1 D<C 1,z E (6) (G(z) G( z)) 1+Dz where G(z) =z + c k z k J(A, B), 1 B<A 1 (7) k=2 In particular, L(1, 1, 1, 1) L J and M(1, 1, 1, 1) M J We study the classes L(A, B, C, D) and M(A, B, C, D) and obtain the coefficient estimates.

3 Coefficient estimate for two subclasses Preliminary Result We need the following preliminary lemma, required for proving our results. Lemma 2.1 If function p(z) = 1+Aω(z) 1+Bω(z) =1+ p k z k,ω(z) U, Then p n A B,n 1 (8) This Lemma is due to Goel and Mehork [2]. Lemma 2.2 If a function f(z) J(A, B), then a n 1 (n 1)(A B) +,n 2 (9) n n Mehrok and Singh proved this result in [3]. 3 Main Results 3.1 Coefficient Estimates for Class L(A, B, C, D) Theorem 3.1 If a function f(z) L(A, B, C, D), then a n 1 n 2 (n 1)(C D)[1 + (A B)] + (n 1)(A B)+1} n2 2 Proof. Setting p(z) = 1+Cω(z) 1+Dω(z) =1+ p k z k,ω(z) U,z E (10) Since f(z) L(A, B, C, D), using (4), we get For (5) and (10) in (11), we have (zf (z)) = H (z)p(z),h(z) J(A, B) (11) (1+4a 2 z+...+k 2 a k z k ) = (1+2b 2 z+...+kb k z k )(1+p 1 z+...+p n z+...) (12) Upon equating coefficients of z n 1 in (12), we have n 2 a n = p n 1 +2b 2 p n (n 1)b n 1 p 1 + nb n

4 384 Liangpeng Xiong and Xiaoli Liu So n 2 a n p n 1 +2 b 2 p n (n 1) b n 1 p 1 + n b n (13) Applying Lemma 2.1 and Lemma 2.2 in (13), we have n 2 a n (C D)[1 + j b j ]+n b n So j=2 (C D)[ (1 + (A B)(j 1))] + (n 1)(A B)+1 j=2 = (C D)[n 1+(A B) (j 1)] + (n 1)(A B)+1 j=2 (n 1)(n 2) = (C D)[n 1+ (A B)] + (n 1)(A B)+1 2 = (n 1)(C D)[1 + n 2 (A B)] + (n 1)(A B)+1 2 a n 1 n 2 (n 1)(C D)[1 + (A B)] + (n 1)(A B)+1} n2 2 On putting A = C = 1,B = D = 1 in the above theorem, we get the following result about the class L J Corollary 3.2 If a function f(z) M J, then a n 1 n 2 + 2(1 1 n ),n Coefficient Estimates for Class M(A, B, C, D) Theorem 3.3 If a function f(z) M(A, B, C, D), then a2n+1 1 (2n+1) 2 (C D)[n + n(n 1)(A B)] + 2n(A B)+1} a 2n 1 4n 2 (C D)[n + n(n 1)(A B)]} Proof. Setting p(z) = 1+Cω(z) 1+Dω(z) =1+ p k z k,ω(z) U,z E (14) Using (6) and (14) in definition 1.2, it gives 2(zf (z)) =[G(z) G( z)] p(z),g(z) J(A, B) (15)

5 Coefficient estimate for two subclasses 385 For(2) and (7), (15) is equal to (1 + 4a 2 z n 2 a n z n n 2 a 2n z 2n 1 +(2n +1) 2 a 2n+1 z 2n +...) =(1+3c 3 z (2n 1)c 2n 1 z 2n )(1 + p 1 z p n z n +...) (16) Upon equating coefficients of z 2n 1 and z 2n in (16),we have (2n +1) 2 a 2n+1 = p 2n +3c 3 p 2n (2n 1)c 2n 1 p 2 +(2n +1)c 2n+1 (17) 4n 2 a 2n = p 2n 1 +3c 3 p 2n (2n 1)c 2n 1 p 1 Applying Lemma 2.1 and Lemma 2.2 in (17), we have (2n +1) 2 a 2n+1 (C D)[1 + (2j +1) c 2j+1 ]+(2n +1) c 2n+1 and (C D)[1 + (1 + 2(A B)j)] + 2(A B)n +1 = (C D)[n +2(A B) j]+2(a B)n +1 = (C D)[n + n(n 1)(A B)] + 2n(A B) + 1 (18) 4n 2 a 2n (C D)[1 + (2j +1) c 2j+1 ] (C D)[1 + (1 + 2(A B)j)] = (C D)[n +2(A B) j] = (C D)[n + n(n 1)(A B)] (19) So following (18) and (19), it gives a2n+1 1 (2n+1) 2 (C D)[n + n(n 1)(A B)] + 2n(A B)+1} a 2n 1 4n 2 (C D)[n + n(n 1)(A B)]} On putting A = C = 1,B = D = 1 in the above theorem, we get the following result about the class M J

6 386 Liangpeng Xiong and Xiaoli Liu Corollary 3.4 If a function f(z) M J, then a2n+1 1 (2n+1) 2 (4n 2 +2n +1) a 2n 1 1 2n ACKNOWLEDGEMENTS: This authors is very much thankful to all the referees. References [1] H.R. Abdel-Gawad and D.K. Thomas, A subclass of close-to-convex functions,publications De L Institut Mathematique, Nouvelle serie,49(63) (1991): [2] R.M. Goel and B.S. Mehork, A subclass of univalent function, Houstan J.Math.,8(3) (1982): [3] B.S. Mehrok and Gagandeep Singh, A Subclass of Close-to-Convex Functions, Int.Journal of Math. Analysis, 4 (2010): Received: October, 2010