Volume 4,  Issue 4, 2003

Article 72

NON-AUTONOMOUS DIFFERENTIAL SUBORDINATIONS RELATED TO A SECTOR

SUKHJIT SINGH AND SUSHMA GUPTA

SANT LONGOWAL INSTITUTE OF ENGINEERING AND TECHNOLOGY,
LONGOWAL-148106 (PUNJAB), INDIA.
E-Mail: sushmagupta1@yahoo.com

Received 02 April, 2003; Accepted 11 September, 2003.
Communicated by: H.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
$\displaystyle (p(z))^\beta \left[1+\lambda(z)\frac {zp^{\prime}(z)}{p^k(z)}\right]^\alpha \prec \left(\frac{1+z}{1-z}\right)^{\gamma_k},  z\in E,$    
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.
Key words:
Univalent function, Starlike function, Subordination, Differential subordination.

2000 Mathematics Subject Classification:
Primary 30C45, Secondary 30C50.


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