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Some Applications of Srivastava-Attiya Operator to p-Valent Starlike Functions
Journal of Inequalities and Applications volume 2010, Article number: 790730 (2010)
Abstract
We introduce and study some new subclasses of p-valent starlike, convex, close-to-convex, and quasi-convex functions defined by certain Srivastava-Attiya operator. Inclusion relations are established, and integral operator of functions in these subclasses is discussed.
1. Introduction
Let denote the class of functions of the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ1_HTML.gif)
which are analytic and p-valent in the open unit disc Also, let the Hadamard product or (convolution) of two functions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ2_HTML.gif)
be given by
A function is said to be in the class
of p-valent functions of order
if it satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ3_HTML.gif)
we write the class of p-valent starlike in
.
A function is said to be in the class
of p-valent convex functions of order
if it satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ4_HTML.gif)
The class of p-valent convex functions in is denoted by
It follows from (1.3) and (1.4) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ5_HTML.gif)
The classes and
were introduced by Goodman [1]. Furthermore, a function
is said to be p-valent close-to-convex of order
and type
in
if there exists a function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ6_HTML.gif)
We denote this class by The class
was studied by Aouf [2]. We note that
was studied by Libera [3].
A function is called quasi-convex of order
type
if there exists a function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ7_HTML.gif)
where We denote this class by
. Clearly
The generalized Srivastava-Attiya operator
in [4] is introduced by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ8_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ9_HTML.gif)
It is not difficult to see from (1.8) and (1.9) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ10_HTML.gif)
When the operator
is well-known Srivastava-Attiya operator [5].
Using the operator we now introduce the following classes:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ11_HTML.gif)
In this paper, we will establish inclusion relation for these classes and investigate Srivastava-Attiya operator for these classes.
We note that
for we get Jung-Kim-Srivastava ([6, 7]);
for we get the generalized Libera integral operator. [8, 9];
for being any negative integer,
and
, the operator
was studied by S
l
gean [10].
2. Inclusion Relation
In order to prove our main results, we will require the following lemmas.
Lemma 2.1 (see [11]).
Let be regular in U with
. If
attains its maximum value on the circle
at a given point
then
where
is a real number and
Lemma 2.2 (see [12]).
Let and let
be a complex function,
Suppose that
satisfies the following conditions:
is continuous in D,
and
for al
such that
Let be analytic in
, such that
for
If
then
for
Our first inclusion theorem is stated as follows.
Theorem 2.3.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_IEq74_HTML.gif)
for any complex number .
Proof.
Let and set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ12_HTML.gif)
where Using the identity
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ13_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ14_HTML.gif)
Differentiating (2.3), logarithmically with respect to , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ15_HTML.gif)
Now, from the function by taking
in (2.4) as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ16_HTML.gif)
it is easy to see that the function satisfies condition (i) and (ii) of Lemma 2.2, in
. To verify condition (iii), we calculate as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ17_HTML.gif)
where and
Therefore, the function
satisfies the conditions of Lemma 2.2.
This shows that if then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ18_HTML.gif)
if then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ19_HTML.gif)
This completes the proof of Theorem 2.3.
Theorem 2.4.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_IEq88_HTML.gif)
, for any complex number s.
Proof.
Consider the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ20_HTML.gif)
which evidently proves Theorem 2.4.
Theorem 2.5.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_IEq89_HTML.gif)
, for any complex number s.
Proof.
Let Then, there exists a function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ21_HTML.gif)
Taking the function which satisfies
we have
and
Now, put where
Using the identity (2.2) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ22_HTML.gif)
Since and
, we let
, where
thus (2.11) can be written as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ23_HTML.gif)
Consider that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ24_HTML.gif)
Differentiating both sides of (2.13), and multiplying by , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ25_HTML.gif)
Using (2.14) and (2.12), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ26_HTML.gif)
Taking in (2.15), we form the function
as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ27_HTML.gif)
It is not difficult to see that satisfies the conditions (i) and (ii) of Lemma 2.2 in
To verify condition (iii), we proceed as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ28_HTML.gif)
where and
being the functions of
and
and
By putting , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ29_HTML.gif)
Hence, and
The proof of Theorem 2.5 is complete
Theorem 2.6.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_IEq118_HTML.gif)
for any complex number s.
Proof.
Consider the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ30_HTML.gif)
The proof of Theorem 2.6 is complete.
3. Integral Operator
For and
, we recall here the generalized Bernardi-Libera-Livingston integral operator
as follows
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ31_HTML.gif)
The operator when
was studied by Bernardi [13], for
was investigated earlier by Libera [14]. Now, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ32_HTML.gif)
so we get the identity
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ33_HTML.gif)
The following theorems deal with the generalized Bernard-Libera-Livingston integral operator defined by (3.1).
Theorem 3.1.
Let If
, then
Proof.
From (3.3), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ34_HTML.gif)
where is analytic in
,
Using (3.3) and (3.4) we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ35_HTML.gif)
Differentiating (3.5), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ36_HTML.gif)
Now we assume that Otherwise, there exists a point
such that
Then by Lemma 2.1, we have
Putting
and
in (3.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ37_HTML.gif)
which contradicts the hypothesis that
Hence, , for
and it follows (3.4) that
The proof of Theorem 3.1 is complete·
Theorem 3.2.
Let If
then
Proof.
Consider the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ38_HTML.gif)
This completes the proof of Theorem 3.2.
Theorem 3.3.
Let If
then
Proof.
Let Then, by definition, there exists a function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ39_HTML.gif)
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ40_HTML.gif)
where From (3.3) and (3.10), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ41_HTML.gif)
Since then from Theorem 3.1, we have
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ42_HTML.gif)
where Using (3.11), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ43_HTML.gif)
Also, (3.10) can be written as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ44_HTML.gif)
Differentiating both sides, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ45_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ46_HTML.gif)
Now, from (3.13) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ47_HTML.gif)
We form the function by taking
in (3.17) as follows
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ48_HTML.gif)
It is clear that the function defined in
by (3.18) satisfies conditions (i) and (ii) of Lemma 2.2. To verify the condition(iii), we proceed as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ49_HTML.gif)
where and
being the functions of
and
and
By putting , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ50_HTML.gif)
Hence, and
. Thus, we have
The proof of Theorem 3.3 is complete.
Theorem 3.4.
Let If
, then
Proof.
Consider the following:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F790730/MediaObjects/13660_2010_Article_2248_Equ51_HTML.gif)
and the proof of Theorem 3.4 is complete.
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The authors would like to thank the referees of the paper for their helpful suggestions.
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Elrifai, E., Darwish, H. & Ahmed, A. Some Applications of Srivastava-Attiya Operator to p-Valent Starlike Functions. J Inequal Appl 2010, 790730 (2010). https://doi.org/10.1155/2010/790730
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DOI: https://doi.org/10.1155/2010/790730