On Certain Subclasses of Multivalent Functions with Varying Arguments of Coefficients

Shigeyoshi Owa, Mohamed K. Aouf and Hanaa M. Zayed abstract: In this paper we introduce and study the classes VMp,η(λ, α, β) and VNp,η(λ, α, β) of multivalent functions with varying arguments of coefficients. We obtain coefficients inequalities, distortion theorems and extreme points for functions in these classes. Also, we investigate several distortion inequalities involving fractional calculus. Finally, results on partial sums are considerd.


Introduction
Let A(p) denote the class of functions of the form: a p+n z p+n (p ∈ N = {1, 2, ...}), (1.1) which are analytic and multivalent in the open unit disc U = {z : z ∈ C and |z| < 1}. We note that A(1) = A.
We recall some definitions which will be used in our paper.
Definition 1.1. [1]. (i) A function f (z) of the form (1.1) is said to be in the class of β−uniformly multivalent starlike functions, denoted by UST p (α, β), if it satisfies the following condition: (ii) A function f (z) of the form (1.1) is said to be in the class of β−uniformly multivalent convex functions, denoted by UCV p (α, β), if it satisfies the following condition: Using the concept of varying arguments in multivalent functions, we introduce the following subclasses.
Also, we note that:

Coefficient estimates
Unless otherwise mentioned, we assume throughout this paper that 0 ≤ α < p, β ≥ 0, 0 ≤ λ < 1, p ∈ N, z ∈ U and φ λ p,n = Γ(p+n+1)Γ(p−λ+1) Proof. Assume that the condition (2.1) holds, then it is sufficient to show the inequality (1.2) holds. Hence, it suffices to show that This last expression is bounded above by (p − α) if (2.1) holds. Conversely, assume that for some sequence {θ p+n } and a real numbers η such that Letting r → 1 − , we obtain the required result and hence the proof of the inequality (2.1) is completed. Further, we consider a function f (z) given by On Certain Subclasses of Multivalent Functions 5 Then, writing We have Therefore, f (z) ∈ VM p,η (λ, α, β) satisfies the equality in (2.1). Then .
The result is sharp for the function It follows that f (z) ∈ VN p (λ, α, β) if and only if (2.2) holds. Moreover, the equality in (2.2) holds true for This completes the proof of Theorem 2.3. .
The result is sharp for the function e iθp+n z p+n .

Distortion theorems
Theorem 3.1. Let f (z) defined by (1.1) be in the class VM p,η (λ, α, β), then for z ∈ U, we have The result is sharp for the function f (z) given by Proof. It is easy to see from Theorem 2.1 that Making use of (3.3), we have which proves the assertion (3.1). Since the equality in (3.1) is satisfied by f (z) given by (3.2), the proof is thus completed.
Using similar arguments to those in the proof of the Theorem 3.1, we obtain the following theorem.
Theorem 3.2. Let f (z) defined by (1.1) be in the class VN p,η (λ, α, β), then for z ∈ U, we have The result is sharp for the function f (z) given by

Extreme points
Theorem 4.1. Let f (z) defined by (1.1) belongs to the class VM p,η (λ, α, β) with arg(a p+n ) = θ p+n and θ p+n + nη ≡ π (mod 2π) for all n. Also, let f p (z) = z p and Then f (z) is in the class VM p,η (λ, α, β) if and only if can be expressed in the form where µ p+n ≥ 0 and ∞ n=0 µ p+n = 1.
Proof. Assume that Then it follows that which implies that f (z) ∈ VM p,η (λ, α, β). Conversely, assume that the function f (z) defined by (1.1) be in the class VM p,η (λ, α, β). Then Corollary 2.2 gives that Defining µ p+n by  This shows that This completes the proof of Theorem 4.1.
By using similar arguments and analysis to those in the proof of Theorem 4.1, we can derive the following theorem.
Theorem 4.2. Let f (z) defined by (1.1) belongs to the class VN p,η (λ, α, β) with arg(a p+n ) = θ p+n and θ p+n + nη ≡ π (mod 2π) for all n. Also, let f p (z) = z p and Then f (z) is in the class VN p,η (λ, α, β) if and only if can be expressed in the form where µ p+n ≥ 0 and ∞ n=0 µ p+n = 1.
We can obtain the following theorem by using similar arguments to those in the proof of the Theorem 6.1.