Some Coefficient Inequalities for Certain Subclasses of Analytic Functions with respect to k-symmetric Points

Transcription

1 Int. Journal of Math. Analysis, Vol. 4, 2010, no. 35, Some Coefficient Inequalities for Certain Subclasses of Analytic Functions with respect to k-symmetric Points V. G. Shanthi 1, B. Srutha Keerthi 2 and B. Adolf Stephen 3 1 Department of Mathematics S.D.N.B. Vaishnav College for Women Chromepet, Chennai , India vg.shanthi1@gmail.com 2 Department of Applied Mathematics Sri Venkateswara College of Engineering Sriperumbudur, Chennai , India sruthilaya06@yahoo.co.in 3 Department of Mathematics Madras Christian College, Tambaram Chennai , India adolfmcc2003@yahoo.co.in Abstract In the present paper, we introduce two new subclasses B (k) (α) and L (k) (α) of analytic functions with respect to k-symmetric points. Some coefficient inequalities for functions belonging to these classes and their subclasses with positive coefficients are provided. Mathematics Subject Classification: 30C45 Keywords: Analytic functions, coefficient inequality, k-symmetric points 1 Introduction Let A denote the class of functions of the form f(z) =z + a n z n,

2 1746 V. G. Shanthi, B. Srutha Keerthi and B. Adolf Stephen which are analytic in the open unit disk Δ = z C : z < 1. Let M(α) be the subclass of A consisting of functions f(z) which satisfy the inequality zf (z) <α (z Δ), f(z) for some α (α >1) and let N (α) be the subclass of A consisting of functions f(z) which satisfy the inequality 1+ zf (z) <α (z Δ), f (z) for some α (α >1). The classes M(α) and N (α) were introduced and investigated recently by Owa and Nishiwaki [1] (see also Srivastava and Attiya [2]). Motivated by M(α) and N (α), the following two subclasses of analytic functions with respect to k-symmetric points were introduced and some interesting results were obtained by Zhi-Gang Wang et al. [3]. A function f(z) Ais in the class M (k) (α) if zf (z) <α (z Δ), f k (z) where α>1, k 1 is a fixed positive integer and f k (z) is defined by the following equality f k (z) = 1 k 1 ε ν f(ε ν z), (ε k =1;z Δ) (1) k ν=0 And a function f(z) A is in the class N (k) (α) if and only if zf (z) M (k) (α). We now provide the following two classes M (k) (k) 1 (α) and N 1 (α), which are subclasses with positive coefficients of the classes M (k) (α) and N (k) (α) respectively. M (k) 1 f(z) (α) = M (k) (α) :f(z) =z + a n z n, with a n 0(n 2) and N (k) 1 (α) = f(z) N (k) (α) :f(z) =z + a n z n, with a n 0(n 2). The subclasses M (k) (k) 1 (α) and N 1 (α) were introduced and studied by Zhi-Gang Wang et al. [3].

3 Coefficient inequalities 1747 Definition 1.1 A function f(z) Ais in the class B (k) (α) if z (1 ) f (z) <α ( 0,z Δ) [f k (z)] (1 ) where α>1, k 1 is a fixed positive integer and f k (z) is given by (1). Definition 1.2 A function f(z) Ais in the class L (k) (α) if zf (z)+z 2 f (z) <α ( 0,z Δ) f k (z) where α>1, k 1 is a fixed positive integer and f k (z) is given by (1). In the present paper, we shall provide some coefficient inequalities for functions belonging to the classes B (k) (α) and L(k) (α) and their subclasses with positive coefficients. 2 Main esults Theorem 2.1 Let α > 1. If f(z) Asatisfies [(nk +1)+ (nk +1) 2α(1 ) ] a nk+1 + then f(z) B (k) (α). 2n a n 2(α 1) (2) Proof Suppose that f(z) Awith α>1, it suffices to show that, z (1 ) f (z) [f k (z)] (1 ) < z (1 ) f (z) 2α [f k (z)](1 ), (z Δ). Let M be denoted by M = z (1 ) f (z) z (1 ) f (z) 2α[f k (z)] (1 ) = z(1 ) + na n z n z(1 ) + na n z n 2αz (1 ) 2α(1 ) a n b n z (n ) where b n = 1 k 1 ε (n 1)ν,(ε k = 1). k ν=0

4 1748 V. G. Shanthi, B. Srutha Keerthi and B. Adolf Stephen Thus, for z = r<1, we have, M r 1 + [ n a n r n (2α 1)r 1 ] n 2α(1 )b n a n r n < [n + n 2α(1 )b n ] a n 2(α 1) r (3) From the definition of b n, we know 1, n = lk +1 b n = 0, n lk +1 (4) Substituting (4) into inequality (3), we get, M< [(nk +1)+ (nk +1) 2α(1 ) ] a nk+1 + 2n a n 2(α 1) r From (2), we know that M<0. Thus we have, z (1 ) f (z) <α, [f k (z)] (1 ) that is f(z) B (k) (α). This completes the proof of Theorem 2.1. Theorem 2.2 Let α > 1. If f(z) Asatisfies (z Δ), [((nk +1)+(nk + 1)(nk)) + (nk +1)+(nk + 1)(nk) 2α ] a nk+1 + then f(z) L (k) (α). 2[n + n(n 1)] a n 2(α 1) (5)

5 Coefficient inequalities 1749 The proof of Theorem 2.2 is similar to Theorem 2.1, so the details are omitted. Corollary 2.3 By substituting =0in Theorem 2.1 and Theorem 2.2, we have, for α>1, f(z) Asatisfies [(nk +1)+ (nk +1) 2α ] a nk+1 + 2n a n 2(α 1) (6) then f(z) M (k) (α) which was studied by Zhi-Gang Wang et al. [3]. We now provide the necessary and sufficient coefficient conditions for the following two classes B (k),1 (α) and L(k),1 (α), which are subclasses with positive coefficients of the classes B (k) (α) and L(k) (α) respectively. B (k),1 (α) = f(z) B (k) (α) :f(z) =z + a n z n, with a n 0(n 2) and L (k),1 (α) = f(z) L (k) (α) :f(z) =z + a n z n, with a n 0(n 2). Theorem 2.4 Let k 2, 1 <α k +1 and f(z) A, then f(z) B (k),1 (α) if and only if na n α(1 ) a lk+1 α 1 Proof In view of Theorem 2.1, we need only to prove the necessity. Let f(z) =z + a n z n B (k),1 (α), then a n 0 for n 2 and this is equivalent to or equivalently, z (1 ) f (z) <α [f k (z)] (1 ) z (1 ) f (z) [f k (z)] (1 ) < z (1 ) f (z) 2α [f k (z)](1 ) z (1 ) f (z) < z (1 ) f (z) 2α[f k (z)] (1 ).

6 1750 V. G. Shanthi, B. Srutha Keerthi and B. Adolf Stephen Hence, 1+ na n z n 1 < 1+ na n z n 1 2α 2α(1 ) a lk+1 z lk Setting z 1, noting that a n 0 for n 2 and α>1, we have, 1+ na n 2α 1+2α(1 ) a lk+1 na n that is, na n α(1 ) a lk+1 α 1. Hence the proof of Theorem 2.4 is complete. Theorem 2.5 Let k 2, 1 <α k +1 and f(z) A, then f(z) L (k),1 (α) if and only if [n + n(n 1)]a n α a lk+1 α 1 The proof of Theorem 2.5 is similar to Theorem 2.4, so the details are omitted. Corollary 2.6 By substituting =0in Theorem 2.4 and Theorem 2.5, we have for k 2, 1 <α k +1 and f(z) A, then f(z) M (k) 1 (α) if and only if na n α a lk+1 α 1 which was studied by Zhi-Gang Wang et al. [3]. eferences [1] S. Owa and J. Nishiwaki, Coefficient estimates for certain classes of analytic functions, J. Inequal. Pure Appl. Math., 3 (2002), Article 72 (electronic). [2] H.M. Srivastava and A.A. Attiya, Some subordination results associated with certain subclasses of analytic functions, J. Inequal. Pure Appl. Math., 5 (2004), Article 82 (electronic).

7 Coefficient inequalities 1751 [3] Zhi-Gang Wang, Chun-Yi Gao and Shao-Mou Yuan, Some coefficient inequalities for certain subclasses of analytic functions with respect to k- symmetric points, Soochow Journal of Mathematics, Volume 33, No. 2, pp , April eceived: March, 2010