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Univalence of Certain Linear Operators Defined by Hypergeometric Function
Journal of Inequalities and Applications volume 2009, Article number: 807943 (2009)
Abstract
The main object of the present paper is to investigate univalence and starlikeness of certain integral operators, which are defined here by means of hypergeometric functions. Relevant connections of the results presented here with those obtained in earlier works are also pointed out.
1. Introduction and Preliminaries
Let denote the class of all analytic functions
in the unit disk
. For
, a positive integer, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ1_HTML.gif)
with , where
is referred to as the normalized analytic functions in the unit disc. A function
is called starlike in
if
is starlike with respect to the origin. The class of all starlike functions is denoted by
. For
, we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ2_HTML.gif)
and it is called the class of all starlike functions of order . Clearly,
for
. For functions
, given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ3_HTML.gif)
we define the Hadamard product (or convolution) of and
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ4_HTML.gif)
An interesting subclass of (the class of all analytic univalent functions) is denoted by
and is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ5_HTML.gif)
where and
The special case of this class has been studied by Ponnusamy and Vasundhra [1] and Obradović et al. [2].
For a,b,c ∈ C and c≠0,-1,-2,…, the Gussian hypergeometric series F(a,b;c;z) is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ6_HTML.gif)
where and
. It is well-known that
is analytic in
. As a special case of the Euler integral representation for the hypergeometric function, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ7_HTML.gif)
Now by letting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ8_HTML.gif)
it is easily seen that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ9_HTML.gif)
For , Owa and Srivastava [3] introduced the operator
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ10_HTML.gif)
which is extensions involving fractional derivatives and fractional integrals. Using definition of we may write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ11_HTML.gif)
This operator has been studied by Srivastava et al. [4] and Srivastava and Mishra [5].
Also for and
, let us define the function
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ12_HTML.gif)
This operator has been investigated by many authors such as Trimble [6], and Obradović et al. [7].
If we take
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ13_HTML.gif)
then we can rewrite operator defined by (1.11) as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ14_HTML.gif)
From the definition of it is easy to check that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ15_HTML.gif)
For with
for all
we define the transform
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ16_HTML.gif)
where and
Also for with
for all
we define the transform
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ17_HTML.gif)
where and
In this investigation we aim to find conditions on such that
implies that the function
to be starlike. Also we find conditions on
for each
the transforms
and
belong to
and
.
For proving our results we need the following lemmas.
Lemma 1.1 (cf. Hallenbeck and Ruscheweyh [8]).
Let be analytic and convex univalent in the unit disk
with
. Also let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ18_HTML.gif)
be analytic in . If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ19_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ20_HTML.gif)
and is the best dominant of (1.20).
Lemma 1.2 (cf. Ruscheweyh and Stankiewicz [8]).
If are analytic and
are convex functions such that
then
.
Lemma 1.3 (cf. Ruscheweyh and Sheil-Small [9]).
Let and
be univalent convex functions in
. Then the Hadamard product
is also univalent convex in
.
2. Main Results
We follow the method of proof adopted in [1, 10].
Theorem 2.1.
Let n be positive integer with . Also let
and
. If
belongs to
, Then
whenever
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ21_HTML.gif)
Proof.
Let us define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ22_HTML.gif)
Since , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ23_HTML.gif)
where is an analytic function with
and
By Schwarz lemma, we have
. By (2.3), it is easy to check that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ24_HTML.gif)
Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ25_HTML.gif)
We need to show that . To do this, according to a well-known result [9] and (2.5) it suffices to show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ26_HTML.gif)
which is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ27_HTML.gif)
Suppose that denote the class of all Schwarz functions
such that
and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ28_HTML.gif)
then, if
. This observation shows that it suffices to find
. First we notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ29_HTML.gif)
Define by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ30_HTML.gif)
Differentiating with respect to
, weget
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ31_HTML.gif)
Case 1.
Let Then we see that
has its only critical point in the positive real line at
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ32_HTML.gif)
Furthermore, we can see that for
and
for
. Hence
attains its maximum value at
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ33_HTML.gif)
Case 2.
Let , then it is easy to see that
and so
attains its maximum value at
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ34_HTML.gif)
Now the required conclusion follows from (2.13) and (2.14).
By putting in Theorem 2.1 we obtain the following result.
Corollary 2.2.
Let be the positive integer with
. Also let
and
. If
belongs to
, then
whenever
.
Remark 2.3.
Taking in Theorem 2.1 and Corollary 2.2 we get results of [10].
We follow the method ofproofadopted in [11].
Theorem 2.4.
Let with
and the function
with
be univalent convex in
. If
and
defined by (1.8) satisfy the conditions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ35_HTML.gif)
then the transform defined by (1.16) has the following:
(1)
(2)whenever
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ36_HTML.gif)
Proof.
From the definition of we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ37_HTML.gif)
Differentiating shows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ38_HTML.gif)
It is easy to see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ39_HTML.gif)
From (1.9) and (2.19) we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ40_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ41_HTML.gif)
Let us define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ42_HTML.gif)
then is analytic in
, with
and
Combining (2.18) with (2.21), one can obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ43_HTML.gif)
Differentiating yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ44_HTML.gif)
In view of (2.21), (2.23), and (2.24), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ45_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ46_HTML.gif)
Since and
are convex and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ47_HTML.gif)
by using Lemmas 1.2 and 1.3, from (2.26) we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ48_HTML.gif)
It now follows from Lemma 1.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ49_HTML.gif)
Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ50_HTML.gif)
and the result follows from the last subordination and Corollary 2.2.
It is well-known that (see, [12]) if and
, then
is univalent convex function in
. So if we take
in the Theorem 2.4, we obtain the following.
Corollary 2.5.
For and
, let the function
and
defined by (1.8) satisfy the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ51_HTML.gif)
Then the transform defined by (1.16) has the following:
(1)
(2) whenever
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ52_HTML.gif)
By putting on the (1.8), we get
which is evidently convex. So by taking
on Theorem 2.4 we have the following.
Corollary 2.6.
For with
, let the function
and
defined by (1.8) satisfy the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ53_HTML.gif)
Then the transform defined by (1.16) has the following:
(1);
(2)whenever
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ54_HTML.gif)
Remark 2.7.
Taking and
on Corollary 2.6, we get a result of [11].
By putting and
on Theorem 2.10 we obtain the following.
Corollary 2.8.
Let and
with
be univalent convex function in
. Also let
with
and
, satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ55_HTML.gif)
and let be the function which is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ56_HTML.gif)
If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ57_HTML.gif)
then we have the following:
(1)
(2) whenever
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ58_HTML.gif)
Remark 2.9.
We note that if , then
is convex function, and so we can replace
with
in Corollary 2.8 to get other new results.
In [13], Pannusamy and Sahoo have also considered the class for the case
with
Theorem 2.10.
For let
and
defined by (1.13) satisfy the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ59_HTML.gif)
Then the transform defined by (1.17) has the following:
(1)
(2) whenever
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ60_HTML.gif)
Proof.
Let us define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ61_HTML.gif)
then is analytic in
, with
and
Using the same method as on Theorem 2.4 we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ62_HTML.gif)
Since and
are convex,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ63_HTML.gif)
Using Lemmas 1.2 and 1.3, from (2.42) it yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ64_HTML.gif)
It now follows from Lemma 1.1 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ65_HTML.gif)
Therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F807943/MediaObjects/13660_2009_Article_2009_Equ66_HTML.gif)
and the result follows from (2.46) and Corollary 2.2.
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Aghalary, R., Ebadian, A. Univalence of Certain Linear Operators Defined by Hypergeometric Function. J Inequal Appl 2009, 807943 (2009). https://doi.org/10.1155/2009/807943
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DOI: https://doi.org/10.1155/2009/807943