JIPAM

Generalized Integral Operator and Multivalent Functions  
 
  Authors: Khalida Inayat Noor,  
  Keywords: Convolution (Hadamard product), Integral operator, Functions with positive real part, Convex functions.  
  Date Received: 20/02/05  
  Date Accepted: 02/03/05  
  Subject Codes:

Primary 30C45, 30C50.

  
  Editors: Themistocles M. Rassias,  
 
  Abstract:

Let $ mathcal{A}(p) $ be the class of functions $ f: f(z) = z^p + sum _{j=1}^{infty }a_j z^{p+j} $ analytic in the open unit disc $ E.$ Let, for any integer $ n> -p, quad f_{n+p-1}(z) = frac{z^{p}}{(1-z)^{n+p}}. $ We define $ f^{(-1)}_{n+p-1}(z) $ by using convolution $ star $ as $ f_{n+p-1}(z) star f_{n+p-1}^{(-1)}(z) = frac{z^{P}}{(1-z)^{n+p} }. $ A function $ p, $ analytic in $ E $ with $ p(0) = 1, $ is in the class $ P_{k}(rho ) $ if $ int _{0}^{2pi }leftvert frac{Re p(z) -rho }{p-rho }rightvert dtheta leq kpi, $ where $ z = re^{ i theta }, k geq 2 $ and $ 0 leq rho p. $ We use the class $ P_{k}(rho ) $ to introduce a new class of multivalent analytic functions and define an integral operator $ quad I_{n+p-1}(f) = f^{(-1)}_{n+p-1} star f(z) $ for $ f(z) $ belonging to this class. We derive some interesting properties of this generalized integral operator which include inclusion results and radius problems.;


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