CONIC REGIONS AND SYMMETRIC POINTS

In this note, the concept of N-symmetric points. Janowski func- tions and the conic regions are combined to define a class of functions in a new interesting domain which represents the conic type regions. certain interesting coefficient inequalities are deduced.


Introduction, Definitions and Preliminaries
Let A denote the class of functions of form Definition 1.1.For two functions f and g analytic in U , we say that the function f is subordinate to g in U and write f (z) ≺ g(z), if there exists a Schwarz function w, which is analytic in U with w(0) = 0 and |w(z)| < 1, such that f (z) = g(w(z)) , z ∈ U .If g is univalent in U then f ≺ g if and only if f (0) = g(0) and f (U ) ⊂ g(U ).
Using the principle of the subordination to define the class P of functions with positive real part [4].Definition 1.2.Let P denote the class of analytic functions of the form p(z) = 1+ ∞ n=1 p n z n defined on U and satisfying p(0) = 1, ℜ{p(z)} > 0, z ∈ U .Any function p in P has the representation p(z) = 1+w(z) 1−w(z) where w(0) = 0, |w(z)| < 1 on U , the class of functions with positive real P plays a crucial role in geometric function theory.Its significance can be seen from the fact that simple subclasses like class of starlike S * , class of convex functions C , class of starlike functions with respect to symmetric points have been defined by using the concept of class of functions with positive real part.The definition of starlike functions with respect to N -symmetric points is as follows.
be its N -weighed mean function.A function f in A is said to belong to the class S * N if functions starlike with respect to N -symmetric points if for every r close to 1 , r < 1, the angular velocity of f about the point M f N (z 0 ) positive at z = z 0 as z traverses the circle |z| = r in the positive direction, that is where If f (z) defined by (1.1) then, where where −1 ≤ B < A ≤ 1, denote the class of analytic function p defined on U with the representation p(z) 1+Bz .Geometrically, a function p(z) ∈ P [A, B] maps the opine unit onto the disk defined by the domain, The class P [A, B] is connected the class P of functions with positive real part by the relation, This class was introduced by Janowski [1] and then studied by several authors.Kanas and Wisniowska [6,11] introduced and studied the class k − U CV of k-uniformly convex functions and the corresponding class k − ST of k-starlike functions.These classes were defined subject to the conic region Ω k , k ≥ 0 given by This domain represents the right half plane for k = 0, hyperbola for 0 < k < 1, a parabola for k = 1 and ellipse for k > 1 .
The functions p k (z) play the role of extremal functions for these conic regions where where [10] where (1.7) These conic regions are being studied by several authors see [5,12].Following are the definitions of the classes k − U CV and k − ST .
Definition 1.6.A function f ∈ A is said to be in the class k − U CV , if and only if, (zf Shams et al [7] further generalized the classes k − ST and k − U CV to KD(k, α) and SD(k, α) respectively subject to the conic domain Now using the concepts of Janowski functions and the conic regions, we define the following.

Geometrically, the function p ∈ k−P [A, B] takes all values from the domain Ω
The domain Ω k [A, B] retains the conic domain Ω k inside the circular region defined by Ω[A, B]. the impact of Ω[A, B] on the conic domain Ω k changes the original shape of the conic regions .The ends of hyperbola and parabola get closer to each other but never meet anywhere and the ellipse gets the oval shape.
When A → 1, B → −1, the radius of the circular disk defined by Ω[A, B] tends to infinity, consequently the arms of hyperbola and parabola expand and the oval turns into ellipse .we see that Ω k [1, −1] = Ω k , the conic domain defined by Kanas and Wisniowska [11].

Main Results
Theorem 2.1.A function f ∈ A and of the form where −1 ≤ B < A ≤ 1, k ≥ 0 and λ N (n) is defined by (1.5).
Proof.Assuming that (2.1) holds, then it suffices to show that We get The last expression is bounded above by 1, then and this completes the proof.
When N = 1, we have the following known result, proved by Khalida Inayat Noor and Sarfraz Nawaz Malik in [9].
If p k (k) = 1 + δ k z + ....., then we have after suitable simplification Now from (2.8), we have Equating coefficients of z n on both sides, we have This implies that By (2.9), we get (2.10) Now we prove that .11)For this, we use the induction method.
Let the hypothesis be true for n = m.From (2.10), we have From (2.7), we have By the induction hypothesis, we have ) .

Multiplying both sides by |δ
, we have . Which shows that inequality (2.11) is true for n = m + 1. Hence the required result.

. 1 )
which are analytic in the open unit disk Received: March 22, 2014 c 2014 Academic Publications, Ltd. url: www.acadpubl.eu§ Correspondence author U = {z : z ∈ C and |z| < 1}, and S denote the subclass of A consisting of all function which are univalent in U .

Definition 1 . 4 .
[2] A function f in A is univalent and starlike with respect to N -symmetric points, or briefly N -starlike if and only if

Definition 1 . 8 .
A function p is said to be in the class k − P [A, B], if and only if, 1) n+1 δ k z + ....... Now we see that the series ∞ B)δ k z + ..... Now if p(z) = 1 + ∞ n=1 c n z n , then by Lemma 1.11, we get