IJPAM: Volume 13, No. 4 (2004)

SOME NEW SUBCLASSES OF ANALYTIC
FUNCTIONS DEFINED BY A CERTAIN
INTEGRAL OPERATOR

Khalida Inayat Noor
Department of Mathematics and Computer Science
College of Science
United Arab Emirates University
P.O. Box 17551, Al-Ain, UNITED ARAB EMIRATES
e-mail: khalidan@uaeu.ac.ae


Abstract.Let ${\cal A} $ be the class of analytic functions defined on the open unit disc $E. $ An integral operator $I_{\alpha } : {\cal A} \longrightarrow {\cal A} $ is defined using the convolution $\star .$ Let $f_\alpha (z) =
\frac{z}{(1-z)^{\alpha + 1}}, \quad \alpha > -1 $ and let $f^{(-1)}_{\alpha } $ be defined such that $ (f_{\alpha }\star
f^{(-1)}_{\alpha })(z)= \frac{z}{1-z}. $ Then, for $ f\in {\cal A}, \quad
\alpha > -1, $ we define
\begin{multline*}
I_{\alpha }f(z) = (f^{(-1)}_{\alpha }\star f )(z) =
[\frac{z}{...
...}}]^{(-1)} \star f(z)\\ = [z _2F_1(1,1;\alpha +1, z ]\star f(z),
\end{multline*}
where $ { _2F_1 }$ is the hypergeometric function. Using this operator, certain new subclasses of analytic functions are defined and studied. Some inclusion results and radii problems are investigated and it is shown that these classes are closed under convolution with convex functions.

Received: April 14, 2004

AMS Subject Classification: 30C45, 30C50

Key Words and Phrases: convolution, convex functions, Noor integral operator, radius

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 13
Issue: 4