Subclass Of K Uniformly Starlike Functions Associated With Wright Generalized Hypergeometric Functions

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1 P a g e 52 Vol.10 Issue 5(Ver 1.0)September 2010 Subclass Of K Uniformly Starlike Functions Associated With Wright Generalized Hypergeometric Functions G.Murugusundaramoorthy 1, T.Rosy 2 And K.Muthunagai 2 GJSFR Classification F (FOR) ,010106, Abstract- In this paper we consider the class of functions of the form (i)for the functions f in S is said to be k uniformly starlike functions of order if if that are satisfying the condition (ii)for the functions f in S is said to be k uniformly convex functions of order if We obtain coefficient bounds, distortion theorem extreme points of the subclass of starlike functions defined by linear operator. Furthermore, we discuss radius of convexity closure properties. Keywords-Univalent, convex, starlike, uniformly convex, uniformly starlike, Linear operator Mathematics Subject Classification: 30C45. For positive real parameters that I. INTRODUCTION Let S be the class of functions that are analytic in the unit disc with f(0) = 0. Denote by T, the subclass of S consists of functions of the form also denote T 1,the subclass of S consisting of functions of the form The Wright generalized hypergeometric function[20] is defined by If A t = 1(t = 1, 2,..., p) B t = 1(t = 1, 2,..., q)we have the relationship: Let T μ be the subclass of T 1 satisfying Following Goodman [9, 10], Rønning[15] defined two subclasses of S, About.G. Murugusundaramoorthy1, T.Rosy2 And K.Muthunagai2 1 School of Science Humanities, VIT University, Vellore , India. gmsmoorthy@yahoo.com 2 Department of Mathematics, Madras Christian College, Chennai thomas.roasy@gmail.com is the generalized hypergeometric function(see for details[8]) N denotes the set of all positive integers is the Pochhammer symbol (6) By using the generalized hypergeometric function Dziok Srivastava [8] introduced the linear operator. In[4] Dziok Raina extended the linear operator by using (5)

2 Vol.10 Issue 5(Ver 1.0),September 2010 P a g e 53 Wright generalized hypergeometric function. First we define a function The proof of the Theorem 1 is similar to that of Theorem 2.2, in [1], hence we omit the details. Theorem 2-Let f(z) be defined by (1.2 ). Then We observe that, for f(z) of the form(1.1),we have is given by (1.6) is defined by (7) Proof- Since If, for convenience, we write (9) introduced by Dziok Raina[4]. In view of the relationship (1.5) the linear operator(1.7) by setting we are led immediately to the aforementioned Dziok- Srivastava operator which contains, as its further special cases, such other linear operators of Geometric Function Theory as the Hohlov operator, the Carlson-Shaffer operator[3], the Ruscheweyh derivative operator[16], the generalized Bernardi-Libera-Livingston operator[2], the fractional derivative operator[7], so on (see, for the precise relationships,dziok Srivastava [4, 5, 6, 8]). For we let denote the subclass of starlike functions corresponding to the family UCV for functions f(z) of the form (??) such that For we let (8) (10) the subclass of Tμ consisting of functions of the form (1.2) satisfying the analytic criterion(1.3). Using the techniques of Silverman [17] motivated by the earlier works [12, 11, 14] [19], in this paper we obtain the coefficient bounds, distortion bounds, extreme points, radius of starlikeness closure theorems for the functions belong to the class Substituting for in (2.1) we get (2.2). (a) Corollary 1-Let the function f(z) defined by (1.2 ) belongs Theorem 3-Let Then the form with equality for it can be expressed in Proof-The proof of the Theorem 3, follows on line similar to the proof of the theorem on extreme points given in Silverman [17] (b) III. A DISTORTION THEOREM Theorem 4-Let the function f(z) defined by (1.2 ) belong to II. MAIN RESULTS Theorem 1. A function f(z) of the form (1.2 ) is in the class

3 P a g e 54 Vol.10 Issue 5(Ver 1.0)September 2010 is non- Proof-In the view of (2.1) the fact that decreasing for we have Therefore, we have which is equivalent to, Using (1.2 ) (3.4), we obtain Which implies that h complete. so the proof is V. RADIUS OF CONVEXITY AND STARLIKENESS In this section we obtain the radius of starlikeness of order radius of convexity of order for the class IV. CLOSURE THEOREMS Let the functions fj(z) be defined for j = 1, 2,...m by (5) Theorem 6-Let Then 1) f is starlike of order in the disc that is, Theorem 5. Let fj(z) defined by (4.1) be in the class Then the function h(z) defined by 2) f is convex of order in the unit disc that is Re also in the same class Each of these results are sharp for the extremal function f(z) given by (2.4). Proof-Given f is starlike of order we have Proof-From (4.2) we have For the left h side of (5.1) we have The last expression is less than if Since applying Theorem 2, we get by Substituting in (5.2), we have

4 Vol.10 Issue 5(Ver 1.0),September 2010 P a g e 55 Using the fact, that We can say (5.1) is true if of the form (1.2 ) is in is in then for we shall prove that there exists a is in Or, equivalently, which yields the starlikeness of the family. Using the fact that f is convex zf0 is starlike, we can prove, on lines similar the proof of. (a) Remark-We note that the radius of starlikeness convexity are independent of the fixed point z0. VI. CONVEX FAMILIES Suppose B is nonempty subset of the real interval (0, 1),we define we observe that f(z) is real when z is real with Hence for some z 1, z 0 z 2 z 1, we have t(z 2 ) = 1. Since z 1, z 2 are arbitrary, the family Conversely, suppose B is not connected. Then we can take z0, z1 z0, z1 2 B, z 2 /2 B such that z 0 < z 2 < z 1. Let us assume f(z) g(z) are not both identity function. Then using (6.1) fixing z = z 2 allow to vary, (6) If B consists of a single element say z0 then is a convex family. Because if f1(z) f2(z) are in then it can be seen that for is in To examine this class for other subsets of B, we prove the following lemma Lemma 1.- z0 z1 are distinct positive numbers then f(z) = z. Since there must exists for which Hence Proof-For the functions of the form (1.2 ), we have Since from the Lemma 1, it follows that That is, Hence so the results follows. (a) Theorem 7.- If B is contained in the interval (0, 1) is a convex family if only if B is connected. Proof-Let B be connected. Suppose Therefore is not a convex family. (b) Concluding Remarks Observe that, if A t = 1(t = 1, 2,..., p) B t = 1(t = 1, 2,..., q) specializing the parameters p, q, 1, 2,..., p, in the class we obtain various classes introduced studied in the literature ( see [12, 14, 17, 19]). VII. REFERENCES 1) K. Aouf G.Murugusundaramoorthy, On a subclass of uniformly convex functions defined by the Dziok-Srivastava Operator, Austral. J.Math.anal. appl., 1 (2008),

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