Study of Multivalent Spirallike Bazilevic Functions

: In this paper, we introduce certain new subclasses of multivalent spirallike Bazilevic functions by using the concept of k -uniformly starlikness and k -uniformly convexity. We prove inclusion relations, su ﬃ cient condition and Fekete-Szego inequality for these classes of functions. Convolution properties for these classes are also discussed.


Introduction
In particular, we write A(1) = A.
Furthermore, by S ⊂ A we shall denote the class of all functions which are univalent in E.
The familiar class of p-valently starlike functions in E, will be denoted by S * (p) which consists of function f ∈ A(p) that satisfy the following conditions One can easily see that where S * is the well-known class of starlike functions. Moreover, for two functions f and g analytic in E, we say that the function f is subordinate to the function g and write as f ≺ g or f (z) ≺ g (z) , if there exists a Schwarz function w which is analytic in E with w (0) = 0 and |w (z)| < 1, such that f (z) = g (w (z)) .
Furthermore, if the function g is univalent in E then it follows that Next, for a function f ∈ A (p) given by (1.1) and another function f ∈ A (p) given by the convolution (or the Hadamard product) of f and g is given by Moreover, the subclass of A consisting of all analytic functions and has positive real part in E is denoted by P. An analytic description of P is given by

Furthermore, if
Re {h(z)} > ρ, then we say that h be in the class P (ρ) . Clearly, one can easily observed that Historically in 1955, Bazilevic [2] define the class of Bazilevic functions, which is the subclass of S, firstly, as follows.
Definition 1.1. For h ∈ P, g ∈ S * and f be given by (1.1) may be represented as where α and γ are real numbers with α > 0. The class of all such Bazilevic functions of type γ is denoted by B(α, γ, h, g).

Definition 1.2. A function f ∈ A is said to be in the class S * (β) if and only if
where R is the set of real numbers.
where R is the set of real numbers.
In fact, Kanas and Wiśniowska were the first (see [7,8]) who defined the conic domain Ω k , k ≥ 0, as and subjected to this domain they also introduced and studied the corresponding class k-ST of kstarlike functions (see Definition 1.4 below). Moreover for fixed k, Ω k represent the conic region bounded successively by the imaginary axis (k = 0) , for k = 1 a parabola, for 0 < k < 1 the right branch of hyperbola and for k > 1 an ellipse. For these conic regions, following functions p k (z), which are given by (1.3) , play the role of extremal functions.
Here K(κ) is Legendre's complete elliptic integral of first kind and Then it was showed in [5] that for (1.3) one can have and with A = 2 π arccos k. These conic regions are being studied and generalized by several authors, for example see [6,15,18].
The class k-ST is define as follows.
In the recent years, several interesting subclasses of analytic functions have been introduced and investigated from different viewpoints for example see ( [1,10,11,13,14,16]). Motivated and inspired by the recent research going on and the above mention work, we here introduce and investigate two new subclasses of analytic functions using the concept of Bazilevic and spirallike functions as follows.
and R is the set of real numbers.

A Set of Lemmas
Each of the following lemmas will be needed in our present investigation. such that for some m ≥ 1.

Main results and their demonstrations
In this section, we will prove our main results.
Theorem 3.1. Let the function be defined by (1.1) and 0 ≤ k < ∞ be a fixed number. If the function f is a member of the function class k-M(β, λ, µ) then for given by (1.4), and (1.6), respectively.
Proof. If f (z) ∈ k-M(β, λ, µ) then there exists a Schwarz function w in E, such that L (β, µ, k, λ) = p k (w (z)). (3.3) where L (β, µ, k, λ) is given by (1.7) . We find after some simplification that where v is given by (3.2) . Making use of (3.4) and (3.5) , we have Taking the moduli in (3.6), we thus obtain In order to prove the first inequality in (3.1), we assume that v > η 1 , then using the estimate from Lemma 2.2 and the known estimate |c 1 | ≤ 1 of the Schwarz Lemma, as a consequence, we have and thus the first inequality in (3.1) is now proved.
To prove the last inequality in the (3.1), for this let v < η 2 , then from (3.7), we have Applying the estimates |c 2 | ≤ 1 − c 2 1 of Lemma 2.2 and the known estimate |c 1 | ≤ 1, we have . This is the last expression of (3.1).