eu SOME PROPERTIES OF STARLIKE AND CLOSE TO CONVEX FUNCTIONS WITH RESPECT TO SYMMETRIC CONJUGATE POINTS

In the present paper, we introduce and investigate some new subclasses of �- starlike and � - close to convex functions with respect to sym- metric conjugate points. Inclusion relationships, integral representations and some interesting convolution properties for these functions are obtained.


Introduction, Definitions and Preliminaries
Let A p be the class of functions f (z), of the form url: www.acadpubl.eu§ Correspondence author functions which are respectively starlike, convex, close-to-convex and quasiconvex in U.The references of the field are [5,6], it covers most of the topics.
Let S be the subclass of A consisting of all functions which are univalent in U. Also, let P denote the class of functions of the form which are analytic and convex in U and satisfy the condition Let f (z) and g(z) be analytic in U. Then we say that the function f (z) is subordinate to g(z) in U, if there exists an analytic function w(z) in U such that |w(z)| < |z| and f (z) = g(w(z)), denoted by f (z) ≺ g(z).If g(z) is univalent in U, then the subordination is equivalent to f (0) = g(0) and f (U) ⊂ g(U).
Let k be a positive integer and j = 0, 1, 2, . . .(k − 1).A domain D is said to be (j, k)-fold symmetric if a rotation of D about the origin through an angle 2πj/k carries where ε = exp(2πi/k).The family of (j, k)-symmetrical functions will be denoted by F j k .For every function f defined on a symmetrical subset U of C, there exits a unique sequence of (j, k)-symmetrical functions f j, k (z), j = 0, 1, . . ., k − 1 such that This decomposition is a generalization of the well known fact that each function defined on a symmetrical subset U of C can be uniquely represented as the sum of an even function and an odd functions (see Theorem 1 of [7]).It is obvious that f j, k (z) is a linear operator from U into U.The notion of (j, k)-symmetrical functions was first introduced and studied by P. Liczberski and J. Polubiński in [7].The class of (j, k)-symmetrical functions was extended to the class (j, k)-symmetrical conjugate functions in [12].For fixed positive integers j and k, let f 2j, k (z) be defined by the following equality If ν is an integer, then the following identities follow directly from (1.4): and ( Motivated by the concept introduced by Sakaguchi in [11], recently several subclasses of analytic functions with respect to k-symmetric points were introduced and studied by various authors (see [1,2,14,15,16]).Parvatham in ( [10]) introduced and investigated K n (α, h) -so called class of α starlike functions with respect to n symmetric points.
Definition 1.1.The function f ∈ A p and f (z) f ′ (z) z = 0 in U is said to be in the class S j, k p (α, φ) if and only if it satisfies the condition where, φ ∈ P, 0 ≤ α ≤ 1 and f 2j, k (z) = 0 is defined by the equality (1.4).Similarly, we say that a function where, φ ∈ P and α ≥ 0.

If we set
then (2.9) can be written as It is very clear that p (z) ≺ φ (z), since α is a real number with the condition imposed 0 ≤ α ≤ 1. Setting . (2.11) Now, by applying the above proof for p (z) ≺ φ (z) and using Lemma 1.2 in (2.11), we know that which implies that S j, k p (α, φ) ⊂ S j, k p (φ).
And also By means of Lemma 1.4 and making use of similar arguments given in the proof for Theorem 2.1, we easily get the following inclusion relationship for the class C j, k p (α, φ).Corollary 2.2.Let φ ∈ P and 0 ≤ α ≤ 1, then Theorem 2.3.Let φ ∈ P and 0 ≤ α ≤ 1, then and , we have, Here q(z) ≺ φ(z)(by lemma).Again an application of Lemma (1.2) yields p (z) = zf ′ (z) pg 2j,k (z) which estabilish the theorem.

Integral Representation
Theorem 3.1.Let f ∈ S j, k p (α, φ) with 0 < α ≤ 1.Then where, f 2j,k (z) is defined by (1.4), w is analytic in U with Proof.Suppose that f ∈ S j, k p (α, φ).We know that the condition (1.7) can be written as where, w is analytic in U with By similar application of the arguments given in the proof for Theorem 2.1 to (3.2), we get, Integrating this equality, we get or equivalently, Now we can derive (3.1) from (3.5).
Proof.Suppose that f ∈ S j, k p (α, φ).Then by (3.2) and (3.5), Integrating the above equality two times, will give the assertions of the theorem.

Convolution Conditions
Let f, g ∈ A p , where f (z) is given by (1.1) and g (z) is defined by then the Hadamard product (or convolution) f * g is defined by We now derive some convolution properties for the function classes S j, k p (α, φ) and C j, k p (α, φ).

1 )
which are analytic in the unit disc U = {z ∈ C : |z| < 1}.And let A=A 1 .We denote by S * , C, K and C * the familiar subclasses of A consisting of Received: August 30, 2014 c 2014 Academic Publications, Ltd.