Non-autonomous Differential Subordinations Related to a Sector

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

and let

be a function analytic in

with

and $p(z) \neq 0$ in

. 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

$\displaystyle p(z)\prec \left(\frac {1+z}{1-z}\right)^\delta$

. 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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