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Some Properties of Certain Class of Integral Operators
Journal of Inequalities and Applications volume 2011, Article number: 531540 (2011)
Abstract
The main object of this paper is to derive some inequality properties and convolution properties of certain class of integral operators defined on the space of meromorphic functions.
1. Introduction and Preliminaries
Let denote the class of functions of the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ1_HTML.gif)
which are analytic in the punctured open unit disk
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ2_HTML.gif)
Let , where
is given by (1.1) and
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ3_HTML.gif)
Then the Hadamard product (or convolution) of the functions
and
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ4_HTML.gif)
For two functions and
, analytic in
, we say that the function
is subordinate to
in
and write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ5_HTML.gif)
if there exists a Schwarz function , which is analytic in
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ6_HTML.gif)
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ7_HTML.gif)
Indeed, it is known that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ8_HTML.gif)
Furthermore, if the function is univalent in
, then we have the following equivalence:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ9_HTML.gif)
Analogous to the integral operator defined by Jung et al. [1], Lashin [2] recently introduced and investigated the integral operator
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ10_HTML.gif)
defined, in terms of the familiar Gamma function, by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ11_HTML.gif)
By setting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ12_HTML.gif)
we define a new function in terms of the Hadamard product (or convolution)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ13_HTML.gif)
Then, motivated essentially by the operator , Wang et al. [3] introduced the operator
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ14_HTML.gif)
which is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ15_HTML.gif)
where (and throughout this paper unless otherwise mentioned) the parameters , and
are constrained as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ16_HTML.gif)
and is the Pochhammer symbol defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ17_HTML.gif)
Clearly, we know that .
It is readily verified from (1.15) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ18_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ19_HTML.gif)
Recently, Wang et al. [3] obtained several inclusion relationships and integral-preserving properties associated with some subclasses involving the operator , some subordination and superordination results involving the operator are also derived. Furthermore, Sun et al. [4] investigated several other subordination and superordination results for the operator
.
In order to derive our main results, we need the following lemmas.
Lemma 1.1 (see [5]).
Let be analytic and convex univalent in
with
. Suppose also that
is analytic in
with
. If
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ20_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ21_HTML.gif)
and is the best dominant of (1.20).
Let denote the class of functions of the form
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ22_HTML.gif)
which are analytic in and satisfy the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ23_HTML.gif)
Lemma 1.2 (see [6]).
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ24_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ25_HTML.gif)
The result is the best possible.
Lemma 1.3 (see [7]).
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ26_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ27_HTML.gif)
In the present paper, we aim at proving some inequality properties and convolution properties of the integral operator .
2. Main Results
Our first main result is given by Theorem 2.1 below.
Theorem 2.1.
Let and
. If
satisfies the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ28_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ29_HTML.gif)
The result is sharp.
Proof.
Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ30_HTML.gif)
Then is analytic in
with
. Combining (1.18) and (2.3), we find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ31_HTML.gif)
From (2.1), (2.3), and (2.4), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ32_HTML.gif)
By Lemma 1.1, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ33_HTML.gif)
or equivalently,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ34_HTML.gif)
where is analytic in
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ35_HTML.gif)
Since and
, we deduce from (2.7) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ36_HTML.gif)
By noting that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ37_HTML.gif)
the assertion (2.2) of Theorem 2.1 follows immediately from (2.9) and (2.10).
To show the sharpness of (2.2), we consider the function defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ38_HTML.gif)
For the function defined by (2.11), we easily find that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ39_HTML.gif)
it follows from (2.12) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ40_HTML.gif)
This evidently completes the proof of Theorem 2.1.
In view of (1.19), by similarly applying the method of proof of Theorem 2.1, we get the following result.
Corollary 2.2.
Let and
. If
satisfies the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ41_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ42_HTML.gif)
The result is sharp.
For the function given by (1.1), we here recall the integral operator
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ43_HTML.gif)
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ44_HTML.gif)
Theorem 2.3.
Let ,
and
. Suppose also that
is given by (2.17). If
satisfies the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ45_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ46_HTML.gif)
The result is sharp.
Proof.
We easily find from (2.17) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ47_HTML.gif)
Suppose that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ48_HTML.gif)
It follows from (2.18), (2.20) and (2.21) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ49_HTML.gif)
The remainder of the proof of Theorem 2.3 is much akin to that of Theorem 2.1, we therefore choose to omit the analogous details involved.
Theorem 2.4.
Let and
. If
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ50_HTML.gif)
and each of the functions satisfies the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ51_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ52_HTML.gif)
The result is sharp when .
Proof.
Suppose that satisfy conditions (2.24). By setting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ53_HTML.gif)
it follows from (2.24) and (2.26) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ54_HTML.gif)
Combining (1.18) and (2.26), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ55_HTML.gif)
For the function given by (2.23), we find from (2.28) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ56_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ57_HTML.gif)
By noting that and
, it follows from Lemma 1.2 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ58_HTML.gif)
Furthermore, by Lemma 1.3, we know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ59_HTML.gif)
In view of (2.24), (2.30), and (2.32), we deduce that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ60_HTML.gif)
When , we consider the functions
which satisfy conditions (2.24) and are given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ61_HTML.gif)
It follows from (2.26), (2.28), (2.30), and (2.34) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ62_HTML.gif)
Thus, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ63_HTML.gif)
The proof of Theorem 2.4 is evidently completed.
With the aid of (1.19), by applying the similar method of the proof of Theorem 2.4, we obtain the following result.
Corollary 2.5.
Let and
. If
is defined by (2.23) and each of the functions
satisfies the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ64_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F531540/MediaObjects/13660_2010_Article_2347_Equ65_HTML.gif)
The result is sharp when .
References
Jung IB, Kim YC, Srivastava HM: The Hardy space of analytic functions associated with certain one-parameter families of integral operators. Journal of Mathematical Analysis and Applications 1993,176(1):138–147. 10.1006/jmaa.1993.1204
Lashin AY: On certain subclasses of meromorphic functions associated with certain integral operators. Computers & Mathematics with Applications 2010,59(1):524–531. 10.1016/j.camwa.2009.06.015
Wang Z-G, Liu Z-H, Sun Y: Some subclasses of meromorphic functions associated with a family of integral operators. Journal of Inequalities and Applications 2009, 2009:-18.
Sun Y, Kuang W-P, Liu Z-H: Subordination and superordination results for the family of Jung-Kim- Srivastava integral operators. Filomat 2010, 24: 69–85. 10.2298/FIL1001069S
Miller SS, Mocanu PT: Differential subordinations and univalent functions. The Michigan Mathematical Journal 1981,28(2):157–172.
Stankiewicz J, Stankiewicz Z: Some applications of the Hadamard convolution in the theory of functions. Annales Universitatis Mariae Curie-Skłodowska Sectio A 1986, 40: 251–265.
Srivastava HM, Owa S (Eds): Current Topics in Analytic Function Theory. World Scientific, River Edge, NJ, USA; 1992:xiv+456.
Acknowledgments
This work was supported by the National Natural Science Foundation under Grant 11026205, the Science Research Fund of Guangdong Provincial Education Department under Grant LYM08101, the Natural Science Foundation of Guangdong Province under Grant 10452800001004255, and the Excellent Youth Foundation of Educational Committee of Hunan Province under Grant 10B002 of the People's Republic of China.
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Zhou, JR., Liu, ZH. & Wang, ZG. Some Properties of Certain Class of Integral Operators. J Inequal Appl 2011, 531540 (2011). https://doi.org/10.1155/2011/531540
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DOI: https://doi.org/10.1155/2011/531540