A subclass of analytic functions

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1 Stud. Univ. Babeş-Bolyai Math. 57(2012), No. 2, A subclass of analytic functions Andreea-Elena Tudor Abstract. In the present paper, by means of Carlson-Shaffer operator and a multiplier transformation, we consider a new class of analytic functions BL(m, l, a, c, µ, α, λ). A sufficient condition for functions to be in this class and the angular estimates are provided. Mathematics Subject Classification (2010): 30C45. Keywords: Analytic functions, Carlson-Shaffer operator, Linear operator. 1. Introduction Let A n denote the class of functions f of the form f() = + a k k, U (1.1) k=n+1 which are analytic in the open unit disc U = { C : < 1}. Let H(U) be the space of holomorphic functions in U. For a C and n N, we denote by H[a, n] = { f H(U), f() = a + a n n + a n+1 n , U } Let the function φ(a, c; ) be given by (a) k φ(a, c; ) := k+1 (c 0, 1, 2,...; U) (c) k k=0 where (x) k is the Pochhammer symbol defined by { 1, k = 0 (x) k := x(x + 1)(x + 2)...(x + k 1), k N Carlson and Shaffer[1] introduced a linear operator L(a, c), corresponding to the function φ(a, c; ), defined by the following Hadamard product: (a) k L(a, c) := φ(a, c; ) f() = + a k+1 k+1 (1.2) (c) k k=1

2 278 Andreea-Elena Tudor We note that (L(a, c)f()) = al(a + 1, c)f() (a 1)L(a, c)f() For a = m + 1 and c = 1 we obtain the Ruscheweyh derivative of f, (see [9]). In [2], N.E. Cho and H. M. Srivastava introduced a linear operator of the form: ( ) m l + k I(m, l)f() = + a k k, m Z, l 0 (1.3) l + 1 We note that k=2 (I(m, l)f()) = (l + 1)I(m + 1, l)f() li(m, l)f() For l = 0 we obtain Sălăgean operator introduced in [10]. Let now L(m, l, a, c, λ) be the operator defined by: L(m, l, a, c, λ)f() = λi(m, l)f() + (1 λ)l(a, c)f() For λ = 0 we get Carlson-Shaffer operator introduced in [1], for λ = 1 we get linear operator in [2] and for a = m + 1, c = 1, l = 0 we get generalied Sălăgean and Ruscheweyh operator introduced by A. Alb Lupaş in [4] By means of operator L(m, l, a, c, λ) we introduce the following subclass of analytic functions: Definition 1.1. We say that a function f A n is in the class BL(m, l, a, c, µ, α, λ), n, m N, l, µ, λ 0, α [0, 1) if ( ) µ L(m + 1, l, a + 1, c, λ)f() 1 L(m, l, a, c, λ)f() < 1 α, U. (1.4) Remark 1.2. The class BL(m, l, a, c, µ, α, λ) includes various classes of analytic univalent functions, such as: BL(0, 0, a, c, 1, α, 1) S (α) BL(1, 0, a, c, 1, α, 1) K(α) BL(0, 0, a, c, 0, α, 1) R(α) BL(0, 0, a, c, 2, α, 1) B(α) introduced by Frasin and Darus in [7] BL(0, 0, a, c, µ, α, 1) B(µ, α) introduced by Frasin and Jahangiri in [6] BL(m, 0, a, c, µ, α, 1) BS(m, µ, α) introduced by A. Alb Lupaş and A. Cătaş in [5] BL(m, 0, m + 1, 1, µ, α, 0) BR(m, µ, α) introduced by A. Alb Lupaş and A. Cătaş in [4] BL(m, 0, m + 1, 1, µ, α, λ) BL(m, µ, α, λ) introduced by A. Alb Lupaş and A. Cătaş in [3]. To prove our main result we shall need the following lemmas: Lemma 1.3. [6] Let p be analytic in U, with p(0) = 1, and suppose that ( ) Re 1 + p () > 3α 1 p() 2α, U (1.5) Then Re p() > α for U and α [ 1/2, 1).

3 A subclass of analytic functions 279 Lemma 1.4. [8] Let p() be an analytic function in U, p(0) = 1 and p() 0, U. If there exist a point 0 U such that arg(p()) < π 2 α, < 0, arg(p( 0 )) = π 2 α with 0 < α 1, then we have 0 p ( 0 ) = ikα p( 0 where k 1 ( a + 1 ) 1 when arg(p( 0 )) = π 2 a 2 α, k 1 ( a + 1 ) 1 when arg(p( 0 )) = π 2 a 2 α, p( 0 ) 1/α = ±ai, a > Main results In the first theorem we provide sufficient condition for functions to be in the class BL(m, l, a, c, µ, α, λ) Theorem 2.1. Let f A n, n, m N, l, µ, λ 0, α [1/2, 1). If λ(l+1)i(m+2,l)f() λli(m+1,l)f()+(1 λ)(a+1)l(a+2,c)f() (1 λ)al(a+1,c)f() L(m+1,l,a+1,c,λ) µ λ(l+1)i(m+1,l)f() λli(m,l)f()+(1 λ)al(a+1,c)f() (1 λ)(a 1)L(a,c)f() L(m,l,a,c,λ) +µ 1 + 3α 1, 2α U (2.1) then f BL(m, l, a, c, µ, α, λ) Proof. If we denote by where p() = L(m + 1, l, a + 1, c, λ)f() p() = 1 + p 1 + p , p() H[1, 1], then, after a simple differentiation, we obtain p () p() = ( ) µ (2.2) L(m, l, a, c, λ)f() λ(l + 1)I(m + 2, l)f() λli(m + 1, l)f() + (1 λ)(a + 1)L(a + 2, c)f() L(m + 1, l, a + 1, c, λ) (1 λ)al(a + 1, c)f() λ)al(a + 1, c)f() (1 λ)(a 1)L(a, c)f() µ(1 L(m + 1, l, a + 1, c, λ) L(m, l, a, c, λ) λ(l + 1)I(m + 1, l)f() λli(m, l)f() µ 1 + µ L(m, l, a, c, λ) Using (2.1), we get ( ) Re 1 + p () > 3α 1 p() 2α.

4 280 Andreea-Elena Tudor Thus, from Lemma 1.3, we get { ( ) µ } L(m + 1, l, a + 1, c, λ)f() Re > α. L(m, l, a, c, λ)f() Therefore, f BL(m, l, a, c, µ, α, λ), by Definition 1.1. If we take a = l in Theorem 2.1, we obtain the following corollary: Corollary 2.2. Let f A n, n, m N, l, µ, λ 0, α [1/2, 1). If (l + 1)L(m + 2, l, a + 2, c, λ) L(m + 1, l, a + 1, c, λ) then f BL(m, l, a, c, µ, α, λ) (l + 1)L(m + 1, l, a + 1, c, λ) µ l + µ(l + 1) L(m, l, a, c, λ) 1 + 3α 1 2α, U Next, we prove the following theorem: Theorem 2.3. Let f() A. If f() BL(m, l, l + 1, c, µ, α, λ), then L(m, l, l + 1, c, λ) arg < π 2 α for 0 < α 1 and 2/π arctan(α/(l + 1)) α(µ 1) = 1 Proof. If we denote by where p() = L(m, l, l + 1, c, λ) p() = 1 + p 1 + p , p() H[1, 1], then, after a simple differentiation, we obtain ( L(m + 1, l, l + 2, c, λ)f() L(m, l, l + 1, c, λ)f() Suppose there exists a point 0 U such that Then, from Lemma 1.4, we obtain: ) µ ( ) µ 1 ( 1 = p() l + 1 arg(p()) < π 2 α, < 0, arg(p( 0 )) = π 2 α If arg(p( 0 )) = π/2 α, then ( ( ) µ ) L(m + 1, l, l + 2, c, λ)f(0 ) 0 arg 0 L(m, l, l + 1, c, λ)f( 0 ) ( ) µ 1 ( 1 = arg p ) ( 0 ) = (µ 1) π2 ( p( 0 ) l + 1 p( 0 ) α + arg (µ 1) π 2 α + arctan α l + 1 = π 2 ( 2 π arctan α α(µ 1) l + 1 p ) () p() ) l + 1 ikα ) = π 2

5 A subclass of analytic functions 281 If arg(p( 0 )) = π/2 α, then ( ( ) µ ) L(m + 1, l, l + 2, c, λ)f(0 ) 0 arg π L(m, l, l + 1, c, λ)f( 0 ) 2 0 These contradict the assumption of the theorem. Thus, the function p() satisfy the inequality arg(p()) < π α, U. 2 If we get, in Theorem 2.2, m = l = 0, µ = 2 and λ = 1, we obtain the following corollary, proved by B. A. Frasin and M. Darus in [7]: Corollary 2.4. Let f() A. If f() B(α), then ( ) f() arg < π 2 α, U for some α(0 < α < 1) and (2/π) tan 1 α α = 1. Acknowledgement. This work was possible with the financial support of the Sectoral Operational Programme for Human Resources Development , co-financed by the European Social Fund, under the project number POSDRU/107/1.5/S/76841 with the title Modern Doctoral Studies: Internationaliation and Interdisciplinarity. References [1] Carlson, B.C., Shaffer, D.B., Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal., 15(1984), no. 4, [2] Cho, N.E., Srivastava, H.M., Argument estimates for certain analytic functions defined by a class of multiplier transformation, Math. Comput Modelling, 37(1-2)(2003), [3] Alb Lupaş, A., Cătaş, A., A note on a subclass of analytic functions defined by a generalied Sălăgean and Ruscheweyh operator, General Mathematics, 17(2009), no. 4, [4] Alb Lupaş, A., Some differential subordinations using Sălăgean and Ruscheweyh operators, Proceedings of International Conference on Fundamental Sciences, ICFS 2007, Oradea, [5] Alb Lupaş, A., Cătaş, A., A note on a subclass of analytic functions defined by differential Sălăgean operator, Buletinul Academiei de Ştiinţe a Republicii Moldova, Matematica, 2(60)(2009), [6] Frasin, B.A., Jahangiri, J.M., A new and comprehensive class of analytic functions, Analele Universităţii din Oradea, XV(2008). [7] Frasin, B.A., Darus, M., On certain analytic univalent functions, Internat. J. Math. and Math. Sci., 25(5)(2001), [8] Oaki, S., Nunokawa, M., The Schwarian derivative and univalent functions, Proc. Amer. Math. Soc., 33(1972), [9] Ruscheweyh, S., New criteria for univalent functions, Proc. Amer. Math. Soc., 49(1975), [10] Sălăgean, G., Subclasses of univalent functions, Lecture Notes in Math, Springer Verlag, Berlin, 1013(1983),

6 282 Andreea-Elena Tudor Andreea-Elena Tudor Babeş-Bolyai University Faculty of Mathematics and Computer Sciences 1 Kogălniceanu Street, Cluj-Napoca, Romania tudor andreea