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  Volume 10, Issue 4, Article 98
 
An Application of Hölder's Inequality for Convolutions
    Authors: Junichi Nishiwaki, Shigeyoshi Owa,  
    Keywords: Analytic function, Multivalent starlike, Multivalent convex.  
    Date Received: 30/03/2009  
    Date Accepted: 16/07/2009  
    Subject Codes:

30C45.

  
    Editors: Nak Eun Cho,  
 
    Abstract:

Let $ mathcal{A}_p(n)$ be the class of analytic and multivalent functions $f(z)$ in the open unit disk $ mathbb{U}$. Furthermore, let $ mathcal{S}_p(n, alpha)$ and $ mathcal{T}_p(n, alpha)$ be the subclasses of $ mathcal{A}_p(n)$ consisting of multivalent starlike functions $f(z)$ of order $ alpha$ and multivalent convex functions $f(z)$ of order $ alpha$, respectively. Using the coefficient inequalities for $f(z)$ to be in $ mathcal{S}_p(n, alpha)$ and $ mathcal{T}_p(n, alpha)$, new subclasses $ mathcal{S}_p^*(n, alpha)$ and $ mathcal{T}_p^*(n, alpha)$ are introduced. Applying the Hölder inequality, some interesting properties of generalizations of convolutions (or Hadamard products) for functions $f(z)$ in the classes $ mathcal{S}_p^*(n, alpha)$ and $ mathcal{T}_p^*(n, alpha)$ are considered.

         
       
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