Some Properties of a Class of Analytic Functions

Neng Xu and R. K. Raina abstract: Making use of convolution, we introduce and investigate a certain class of functions which is analytic in the open unit disk. We obtain interesting properties of starlikeness and convexity for this function class. Special cases and some useful consequences of our main results are also mentioned.


Let the functions
Let A be the class of normalized functions φ(z) of the form which are analytic in the open unit disk U.The subclasses of the class A denoted by S * (α) and K(α) are , respectively, the subclasses of starlike functions of order α(0 ≤ α < 1) in U, and the convex functions of order α(0 ≤ α < 1) in U.In Neng Xu and R. K. Raina particular, the classes S * (0) = S * and K(0) = K are well known classes of starlike and convex functions in U, respectively.For functions φ(z) ∈ A and ϕ(z) ∈ A. We say that the function φ(z) is subordinate to ϕ(z) in U, and we write φ(z) ≺ ϕ(z), if there exists an analytic function w(z) in U such that |w(z)| ≤ |z| and φ(z) = ϕ(w(z)) (z ∈ U).

Let the functions
be analytic in the open unit disk U. We introduce and investigate a class of functions Φ λ F,G (z) defined by which is analytic in the open unit disc U, where the functions F and G are of the form (1.2) with the coefficients, a k and b k , respectively, replaced by f k ≥ 0 and g k ≥ 0. By applying elementary calculations, we observe that which asserts that the class Φ(z) ∈ A.
On the other hand, when The class of functions F λ (z) was studied by Raina and Bansal [6] and contains as special cases the classes due to Fukui et al. [2] and Reade at al. [7].In this paper we investigate the geometric properties of starlikeness and convexity for the function class Φ(z) defined above by (1.3).We also consider some relevant particular cases of our main results by mentioning few known (and new) results.

Main results
The starlikeness property satisfied by the class of functions Φ(z) defined by (1.3) is contained in Theorem 1 below.
Proof.In order to prove that Φ(z) Differentiating (1.3) with respect to z, we get
Definition.A function Φ(z) ∈ A given by (1.3) is said to be in the class if and only if it satisfies and the coefficient inequality (2.1).
then it is easy to verify that where We have thus the following inclusion relations: To establish the next result, we state here a known lemma which is due to Ruscheweyh [8].
By applying Lemma 1, we derive following convolution conditions for S * (A, B).
Theorem 5. Let Φ(z) be defined by (1.3), then Φ(z) ∈ K(α)(0 ≤ α < 1), provided that there exist numbers p, q, where 1 p + 1 q ≤ 1, satisfying the following inequalities: Proof.Let the inequalities (2.18) and (2.19) be satisfied for the function Φ(z).We prove that After some calculations, we get where It readily follows that and in view of (2.18), we infer that Making use of (2.19), we get  3.Some consequences of main results In this concluding section, we consider some consequences of our main results proved in section 2.
Remark 3.For and From Theorems 1 and 4, we have Corollary 4. The function Also, from Theorem 5, we have , provided that there exist number p, q > 0 such that 1 p + 1 q ≤ 1 satisfying the following inequalities: where W λ n (z) be defined by (2.24).It is observed that when n = 1, then W λ 1 = F λ (z), where F λ (z) is defined by (1.