A general coefficient inequality 1

Transcription

1 General Mathematics Vol. 17, No. 3 (2009), A general coefficient inequality 1 Sukhwinder Singh, Sushma Gupta and Sukhjit Singh Abstract In the present note, we obtain a general coefficient inequality regarding a multiplier transformation in the open unit disc E = {z : z < 1}. As special case to our main result, we obtain a coefficient inequality for n-starlikeness of analytic functions. Also certain known coefficient inequalities for starlikeness and convexity of analytic functions appear as particular cases of our main result Mathematics Subject Classification: 30C45, 30C50. Key words and phrases: Analytic function, Starlike function, Convex function, Multiplier transformation. 1 Received 4 September, 2008 Accepted for publication (in revised form) 16 September,

2 100 S. Singh, S. Gupta and S. Singh 1 Introduction Let A p denote the class of functions of the form f(z) = z p + a k z k, p N = {1, 2, }, which are analytic in the open unit disc E = {z : z < 1}. A 1 = A. We write Denote by S (α) and K(α), the classes of starlike functions of order α and convex functions of order α respectively, which are analytically defined as follows: and K(α) = S (α) = { f(z) A : R zf } (z) f(z) > α, z E { ( f(z) A : R 1 + zf ) } (z) > α, z E f (z) where α is a real number such that 0 α < 1. We shall use S and K to denote S (0) and K(0), respectively which are the classes of univalent starlike (w.r.t. the origin) and univalent convex functions. as For f A p, we define the multiplier transformation I p (n, λ) as I p (n, λ)f(z) = z p + ( ) n k + λ a k z k, (λ 0, n Z). p + λ I 1 (n, 0) is the well-known Sălăgean [2] derivative operator D n, defined and for f A. D n f(z) = z + k n a k z k, n N 0 = N {0},

3 A general coefficient inequality 101 Denote by S n(α), the class of n-starlike functions of order α, which is analytically defined as follows { Sn(α) = f(z) A : R Dn+1 f(z) D n f(z) where α is a real number such that 0 α < 1. } > α, z E In the present note, we obtain a general coefficient inequality regarding multiplier transformation I p (n, λ) in the open unit disc E = {z : z < 1}. As special case to our main result, we obtain a coefficient inequality for n- starlikeness of analytic functions. Also certain known coefficient inequalities for starlikeness and convexity of analytic functions appear as particular cases of our main result. 2 Main Result Theorem 1 If f A p satisfies p + λ p + λ 1 + p α a k p α, then I p (n + 1, λ)f(z) I p (n, λ)f(z) 1 < p α, 0 α < p, z E. Proof. To prove the required result, we prove that I p (n + 1, λ)f(z) I p (n, λ)f(z) (p α) I p (n, λ)f(z) 0 Indeed, we have I p (n + 1, λ)f(z) I p (n, λ)f(z) (p α) I p (n, λ)f(z)

4 102 S. Singh, S. Gupta and S. Singh = = < p + λ p + λ 1 a k z k ( ) n k + λ (p α) zp + a k z k p + λ p + λ p + λ 1 a k z k ( ( ) n k + λ (p α) z p a k z ) k p + λ p + λ p + λ 1 + p α a k z k (p α) z p p + λ p + λ 1 + p α a k (p α) 0. Hence I p (n + 1, λ)f(z) I p (n, λ)f(z) 1 < p α, 0 α < p, z E. 3 Applications For λ = 0 and p = 1 in Theorem 1, we obtain the following result. Corollary 1 If f A satisfies k n (k α) a k 1 α, then f S n(α). For λ = 0, p = 1 and n = 0 in Theorem 1, we obtain the following result.

5 A general coefficient inequality 103 Corollary 2 If f A satisfies (k α) a k 1 α, then f S (α). For λ = 0, p = 1 and n = 1 in Theorem 1, we obtain the following result. Corollary 3 If f A satisfies k(k α) a k 1 α, then f K(α). Remark 1 For α = 0, Corollary 2 and Corollary 3 are the results of Clunie and Keogh [1] and Silverman [3]. References [1] Clunie, J. and Keogh, F.R., On starlike and convex schlicht functions, J. londan Math. Soc., 35(1960), [2] Sălăgean, G. S., Subclasses of univalent functions, Lecture Notes in Math., 1013, , Springer-Verlag, Heideberg,1983.

6 104 S. Singh, S. Gupta and S. Singh [3] Silverman, H., Univalent functions with negative coefficient, Proc. Amer. Math. Soc., 51(1975), Sukhwinder Singh Department of Applied Sciences Baba Banda Singh Bahadur Engineering College Fatehgarh Sahib (Punjab), INDIA ss billing@yahoo.co.in Sushma Gupta, Sukhjit Singh Department of Mathematics Sant Longowal Institute of Engineering & Technology Longowal (Punjab), INDIA sushmagupta1@yahoo.com, sukhjit d@yahoo.com