Subclasses of starlike functions associated with some hyperbola 1

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1 General Mathematics Vol. 14, No , Subclasses of starlike functions associated with some hyperbola 1 Mugur Acu Abstract In this paper we define some subclasses of starlike functions associated with some hyperbola by using a generalized Sălăgean operator and we give some properties regarding these classes Mathematics Subject Classification: 30C45 Key words and phrases: Starlike functions, Libera-Pascu integral operator, Briot-Bouquet differential subordination, generalized Sălăgean operator 1 Introduction Let HU be the set of functions which are regular in the unit disc U, A = {f HU : f0 = f 0 1 = 0}, H u U = {f HU : f is univalent in U} and S = {f A : f is univalent in U}. 1 Received April 4, 2006 Accepted for publication in revised form April 10,

2 32 Mugur Acu Let D n be the Sălăgean differential operator see [12] defined as: D n : A A, n N and D 0 = D 1 = D = zf z, D n = DD n 1. Remark 1.1. If f S, = a j z j, z U then D n = j n a j z j. 1 We recall here the definition of the well - known class of starlike functions { S = f A : Re zf } z > 0, z U. Let consider the Libera-Pascu integral operator L a : A A defined as: = L a F z = 1 + a z a z 0 F t t a 1 dt, a C, Re a 0. Generalizations of the Libera-Pascu integral operator was studied by many mathematicians such are P.T. Mocanu in [7], E. Drăghici in [6] and D. Breaz in [5]. Definition 1.1.[4] Let n N and 0. We denote with D n defined by D n : A A, the operator D 0 =, D 1 = 1 + zf z = D, D n = D D n 1. Remark 1.2.[4] We observe that D n is a linear operator and for = a j z j we have D n = 1 + j 1 n a j z j.

3 Subclasses of starlike functions associated with some hyperbola 33 Also, it is easy to observe that if we consider = 1 in the above definition we obtain the Sălăgean differential operator. The next theorem is result of the so called admissible functions method introduced by P.T. Mocanu and S.S. Miller see [8], [9], [10]. Theorem 1.1. Let h convex in U and Re[βh γ] > 0, z U. If p HU with p0 = h0 and p satisfied the Briot-Bouquet differential subordination p zp z βp γ hz, then pz hz. In [1] is introduced the following operator: Definition 1.2. Let β, R, β 0, 0 and = denote by D β the linear operator defined by a j z j. We D β : A A, D β = 1 + j 1 β a j z j. Remark 1.3. It is easy to observe that for β = n N we obtain the Al- Oboudi operator D n and for β = n N, = 1 we obtain the Sălăgean operator D n. The purpose of this note is to define some subclasses of starlike functions associated with some hyperbola by using the operator D β defined above and to obtain some properties regarding these classes.

4 34 Mugur Acu 2 Preliminary results Definition 2.1. [13] A function f S is said to be in the class SHα if it satisfies zf z 2α 2 1 < Re for some α α > 0 and for all z U. { } 2 zf z + 2α 2 1, Remark 2.1. Geometric { interpretation: } zf z Let Ωα = : z U, f SHα. Then Ωα = {w = u + i v : v 2 < 4αu + u 2, u > 0}. Note that Ωα is the interior of a hyperbola in the right half-plane which is symmetric about the real axis and has vertex at the origin. Definition 2.2. [3] Let f S and α > 0. We say that the function f is in the class SH n α, n N, if D n+1 2α 2 1 < Re D n { } 2 D n+1 +2α 2 1, z U. D n Remark 2.2. Geometric interpretation: If we denote with p α the analytic and univalent functions with the properties p α 0 = 1, p α0 > 0 and p α U = Ωα see Remark 2.1, then f SH n α if and only if D n+1 p D n α z, where the symbol denotes the subordination in U. 1 + bz 1 + 4α 4α2 We have p α z = 1 + 2α 2α, b = bα = and the 1 z 1 + 2α 2 branch of the square root w is chosen so that Im w 0. Theorem 2.1. [3] If F z SH n α, α > 0, n N, and = L a F z, where L a is the integral operator defined by 1, then SH n α, α > 0, n N.

5 Subclasses of starlike functions associated with some hyperbola 35 Theorem 2.2. [3] Let n N and α > 0. If f SH n+1 α then f SH n α. 3 Main results 1 + bz Definition 3.1. Let β 0, 0, α > 0 and p α z = 1 + 2α 1 z 2α, 1 + 4α 4α2 where b = bα = and the branch of the square root w is chosen so that Im w 0. We say that a function S is in the 1 + 2α 2 class SH β, α if D β p α z, z U. Remark 3.1. Geometric interpretation: SH β, α if and only if D β take all values in the domain Ωα which is the interior of a hyperbola in the right half-plane which is symmetric about the real axis and has vertex at the origin see Remark 2.1 and Remark 2.2. Remark 3.2. It is easy to observe that for β = n N and = 1 we obtain in the above definition we obtain the class SH n α studied in [3] and for = 1, β = 0 we obtain the class SHα studied in [13]. Theorem 3.1. Let β 0, α > 0 and > 0. We have SH β+1, α SH β, α. Proof. Let SH β+1, α. With notation pz = Dβ+1, p0 = 1, D β

6 36 Mugur Acu 2 we obtain Also, we have β+2 = D D β D β = D β = 1 pz Dβ+2 D β 1 + j 1 β+2 a j z j 1 + j 1 β a j z j and or 3 We have z zp z = D β Dβ+1 z D β D β D β = z j 1 β+1 ja j z j 1 = pz z zp z = pz 1 + D β 1 + j 1 β ja j z j 1 D β j 1 + j 1 β+1 a j z j D β j 1 + j 1 β a j z j D β. j 1 + j 1 β+1 a j z j =

7 Subclasses of starlike functions associated with some hyperbola 37 = = = j j 1 β+1 a j z j = 1 + j 1 β+1 a j z j + = = = j j 1 β+1 a j z j = j j 1 β+1 a j z j = j j 1 β+1 a j z j = 1 + j j 1 β+1 a j z j = j 1 β+1 a j z j j 1 β+2 a j z j = = 1 = 1 z = 1 Dβ+1 + z + 1 Dβ+2 z = = 1 = 1 Similarly we have Dβ Dβ+2 1 j 1 + j 1 β a j z j = 1 From 3 we obtain = +. 1D β + Dβ+1. zp z = = D β pz 1Dβ + Dβ+1 D β =

8 38 Mugur Acu Thus or = 1 1p Dβ+2 D β From 2 we obtain where > 0. = 1 pz 1 + pz D β pz 2 zp z = Dβ+2 D β pz 2 D β = pz 2 + zp z. = 1 pz 2 + zp z = p zp z pz pz, From SH β+1, α we have p zp z pz p αz, with p0 = p α 0 = 1, α > 0, β 0, > 0, and Re p α z > 0 from here construction. In this conditions from Theorem 1.1, we obtain or This means SH β, α. pz p α z D β p α z. Theorem 3.2. Let β 0, α > 0 and 1. If F z SH β, α then = L a F z SH β, α, where L a is the Libera-Pascu integral operator defined by 1. =

9 Subclasses of starlike functions associated with some hyperbola 39 Proof. From 1 we have 1 + af z = a + zf z and, by using the linear operator, we obtain 1 + a F z = a + = a + ja j z j = 1 + j 1 β+1 ja j z j We have see the proof of the above theorem Thus or j 1 + j 1 β+1 a j z j = a F z = a + 1 = a = + 1 Dβ a F z = a Similarly, we obtain Then 1 + ad β F z = a Dβ + Dβ+1. F z D β F z = β+1 D D β + a Dβ+1 D β D β + a

10 40 Mugur Acu With notation we obtain 4 F z D β F z = D β = pz, p0 = 1, p a pz p a We have see the proof of the above theorem zp z = Dβ+2 β+1 D D β pz 2 = = Dβ+2 pz pz2. Thus = 1 pz pz 2 + zp z. Then, from 4, we obtain. F z D β F z = pz2 + zp a pz p a = zp z = p p a + 1 1, where a C, Re a 0, β 0, and 1. From F z SH β, α we have zp z p 1 p a p αz, where a C, Re a 0, α > 0, β 0, 1, and from her construction, we have Re p α z > 0. In this conditions we have from Theorem 1.1 we obtain pz p α z

11 Subclasses of starlike functions associated with some hyperbola 41 or D β p α z. This means = L a F z SH β, α. Remark 3.3. If we consider β = n N in the previously results we obtain the Theorem 3.1 and Theorem 3.2 from [2]. References [1] M. Acu and S. Owa, Note on a class of starlike functions, Proceedings of the International Short Point Research Work on Study on Calculus Operators in Univalent Function Theory - kyoto 2006 to appear. [2] M. Acu and S. Owa, On n-starlike functions associated with some hyperbola, IJMS to appear. [3] M. Acu, On a subclass of n-starlike functions associated with some hyperbola, General Mathematics, Vol. 13, No , [4] F.M. Al-Oboudi, On univalent funtions defined by a generalized Sălăgean operator, Ind. J. Math. Math. Sci. 2004, no , [5] D. Breaz, Operatori integrali pe spaţii de funcţii univalente, Editura Academiei Române, Bucureşti [6] E. Drăghici, Elemente de teoria funcţiilor cu aplicaţii la operatori integrali univalenţi, Editura Constant, Sibiu [7] P.T. Mocanu, Classes of univalent integral operators, J. Math. Anal. Appl. 157, 11991,

12 42 Mugur Acu [8] S. S. Miller and P. T. Mocanu, Differential subordonations and univalent functions, Mich. Math , [9] S. S. Miller and P. T. Mocanu, Univalent solution of Briot-Bouquet differential equations, J. Differential Equations , [10] S. S. Miller and P. T. Mocanu, On some classes of first-order differential subordinations, Mich. Math , [11] W. Rogosinski, On the coefficients of subordinate functions, Proc. London Math. Soc , [12] Gr. Sălăgean, Subclasses of univalent functions, Complex Analysis. Fifth Roumanian-Finnish Seminar, Lectures Notes in Mathematics, 1013, Springer-Verlag, 1983, [13] J. Stankiewicz and A. Wisniowska, Starlike functions associated with some hyperbola, Folia Scientiarum Universitatis Tehnicae Resoviensis 147, Matematyka , University Lucian Blaga of Sibiu Department of Mathematics Str. Dr. I. Raṭiu, No Sibiu, Romania acu mugur@yahoo.com