Some Applications of a First order Differential Subordination  
  Authors: Sukhjit Singh, Sushma Gupta,  
  Keywords: Univalent function, Starlike function, Convex function, Differential subordination.  
  Date Received: 28/01/04  
  Date Accepted: 19/07/04  
  Subject Codes:

Primary 30C45, Secondary 30C50.

  Editors: Alexandru Lupas (1942-2007),  

Let $ p$ and $ q$ be analytic functions in the unit disc $ E={z: vert zvert1}$, with $ p(0)=q(0)=1$. Assume that $ alpha$ and $ delta$ are real numbers such that $ 0 delta leq 1$, $ alpha +delta geq 0$. Let $ beta$ and $ gamma$ be complex numbers with $ beta neq 0 $. In the present paper, we investigate the differential subordination

$displaystyle {{(p(z))^ alpha} left[pleft (zright)+frac {zp prime (z)} { ... ...t[ q(z)+ frac {zqprime (z)}{ beta q(z)+gamma} right]^delta},quad zin E,$    

and as applications, find several sufficient conditions for starlikeness and univalence of functions analytic in the unit disc $ E$.;

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