Coefficient Inequalities for Uniformly P-Valent Starlike and Convex Functions

Let Kp(α) denote the class of all such functions. For p=1 we write A1=A. Note that for p=1 the classes 1 ( ) α ∗ S and K1(α) are the usual classes of starlike and convex functions of order α(0 ≤ α<1) respectively, and will be denoted by S* (α) and K(α) respectively. For p=1 and α=0, the classes ( ) α p S and Kp(α) reduces to S*(0)=S*and K(0)=K respectively, which are the classes of starlike (with respect to the origin) and convex functions.


Introduction
The class of all such functions is denoted by ( ) α * p S . A function f ∈A p is said to be p−valent convex of order α (0 ≤ α<p), if , z U f ' ( z ) Let K p (α) denote the class of all such functions. For p=1 we write A1=A. Note that for p=1 the classes 1 ( ) α * S and K 1 (α) are the usual classes of starlike and convex functions of order α(0 ≤ α<1) respectively, and will be denoted by S* (α) and K(α) respectively. For p=1 and α=0, the classes ( ) α * p S and Kp(α) reduces to S*(0)=S*and K(0)=K respectively, which are the classes of starlike (with respect to the origin) and convex functions.

The Subclasses SD p (β,α) and KD p (β,α)
We begin this Section by remark that this article is motivated by the work of Owa et al. [1]. We now recall the definitions of the subclasses SD p (β, α) and KD p (β, α) of uniformly p−valent function introduced and studied by Agnihotri and Singh [2].
A function f∈Ap is said to be in the class SD p (β, α) if for some β ≥ 0 and α (0 ≤ α<p).

Coefficient Inequalities
We now give coefficient inequalities for functions belonging to the subclasses SD p (β, α) and KD p (β, α). Our first result is contained in for any complex number z. Therefore .
Next, we state the corresponding result for functions belonging to the subclass Proof is similar to the proof of Theorem 3.1.

Corollary 3.2: If
We now state the main theorem of this paper.

Theorem 3.3: If
Note that q is analytic in U with Thus, (3.4) holds for n=3. Next, we assume that (3.4) is true for n=k and therefore This shows that (3.4) is true for n=k+1. Hence, by using the principle of mathematical induction, (3.4) holds for all n ≥ 2.    which was proven by Robertson [5].
We know that Define a function r(z) by Note that r is analytic in U with r(0)=1 and ( ( )) 0 ℜ > r z If Thus, (3.12) holds for n=3. Next, we assume that (3.12) is true for n=k and therefore  This shows that (3.12) is true for n=k + 1. Hence, by using the principle of mathematical induction, (3.12) holds for all n ≥ 2.