Hindawi Publishing Corporation
ISRN Mathematical Analysis
Volume 2013, Article ID 384170, 8 pages
http://dx.doi.org/10.1155/2013/384170
Research Article
Some Inclusion Relationships of Certain Subclasses of -Valent
Functions Associated with a Family of Integral Operators
M. K. Aouf,
1
R. M. El-Ashwah,
2
and Ahmed M. Abd-Eltawab
3
1
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
2
Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt
3
Department of Mathematics, Faculty of Science, Fayoum University, Fayoum 63514, Egypt
Correspondence should be addressed to Ahmed M. Abd-Eltawab; ams03@fayoum.edu.eg
Received 16 June 2013; Accepted 24 August 2013
Academic Editors: G. Gripenberg and B. Wang
Copyright © 2013 M. K. Aouf et al. Tis is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
By making use of the new integral operator R
,
,
, we introduce and investigate several new subclasses of -valent starlike, -valent
convex, -valent close-to-convex, and -valent quasi-convex functions. In particular, we establish some inclusion relationships
associated with the aforementioned integral operators. Some of the results established in this paper would provide extensions of
those given in earlier works.
1. Introduction
Let () denote the class of functions of the form
()=
+
∞
∑
=1
+
+
(∈ N ={1,2,3,...}), (1)
which are analytic and -valent in the unit disc ={:∈
C, and || < 1} and let (1) = .
A function ∈ () is said to be in the class
∗
() of
-valent starlike functions of order in if and only if
Re (
()
()
)> (∈;0≤<). (2)
Te class
∗
() was introduced by Patil and Takare [1].
Owa [2] introduced the class
() of -valent convex of
order in if and only if
Re (1+
()
()
)> (∈;0≤<). (3)
It is easy to observe from (2) and (3) that
()∈
() ⇐⇒
()
∈
∗
(). (4)
We denote by
∗
=
∗
(0) and
=
(0) where
∗
and
are the classes of -valently starlike functions and -
valently convex functions, respectively, (see Goodman [3]).
For a function ∈ (), we say that ∈
(,) if there
exists a function ∈
∗
() such that
Re (
()
()
)> (∈;0≤,<). (5)
Functions in the class
(,) are called -valent close-to-
convex functions of order and type . Te class
(,)
was studied by Aouf [4] and the class
1
(,) was studied
by Libera [5].
Noor [6, 7] introduced and studied the classes
∗
(,)
and
∗
1
(,) as follows.
A function ∈ () is said to be in the class
∗
(,) of
quasi-convex functions of order and type if there exists a
function ∈
() such that
Re
{
{
{
(
())
()
}
}
}
> (∈;0≤,<). (6)