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  Volume 4, Issue 4, Article 72
 
Non-autonomous Differential Subordinations Related to a Sector

    Authors: Sukhjit Singh, Sushma Gupta,  
    Keywords: Univalent function, Starlike function, Subordination, Differential subordination.  
    Date Received: 02/04/03  
    Date Accepted: 11/09/03  
    Subject Codes:

30C45,30C50.

  
    Editors: Hari M. Srivastava,  
 
    Abstract:

Let $ lambda(z)$ be a complex valued function defined in the unit disc $ E$ and let $ p(z)$ be a function analytic in $ E$ with $ p(0)=1$ and $ p(z) neq 0$ in $ E$. In this article, we determine the largest constants $ gamma_k,k=1,2,3,ldots $ and conditions on $ lambda(z)$ such that for given $ alpha,beta $ and $ delta$, the non-autonomous differential subordination

$displaystyle (p(z))^beta left[1+lambda(z)frac {zp^{prime}(z)}{p^(z)}right]^alpha prec left(frac{1+z}{1-z}right)^{gamma_k},  zin E,$   
implies
$displaystyle p(z)prec left(frac {1+z}{1-z}right)^delta$    
in $ E$. Here the symbol ` $ prec $' stands for subordination. Almost all the previously known results on differential subordination concerning a sector follow as particular cases of our results.

         
       
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