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Certain Classes of Harmonic Multivalent Functions Based on Hadamard Product
Journal of Inequalities and Applications volume 2009, Article number: 759251 (2009)
Abstract
We define and investigate two special subclasses of the class of complex-valued harmonic multivalent functions based on Hadamard product.
1. Introduction
A continuous function is a complex-valued harmonic function in a complex domain
if both
and
are real harmonic in
. In any simply connected domain
, we can write
, where
and
are analytic in
. We call
the analytic part and
the co-analytic part of
. Note that
reduces to
if the coanalytic part
is zero.
For , denote by
the set of all multivalent harmonic functions
defined in the open unit disc
, where
and
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ1_HTML.gif)
are analytic functions in .
Let be a fixed multivalent harmonic function given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ2_HTML.gif)
A function is said to be in the class
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ3_HTML.gif)
where is a harmonic convolution of
and
. Note that
. Using the fact
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ4_HTML.gif)
it follows that if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ5_HTML.gif)
A function in
is called
-uniformly multivalent harmonic starlike function associated with a fixed multivalent harmonic function
. The set
is a comprehensive family that contains several previously studied subclasses of
; for example, if we let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ6_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ7_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ8_HTML.gif)
(see [3]);
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ9_HTML.gif)
(see [4]);
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ10_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ11_HTML.gif)
(see [7]);
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ12_HTML.gif)
(see [8]).
Finally, denote by the subclass of functions
in
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ13_HTML.gif)
Let .
In this paper, we investigate coefficient conditions, extreme points, and distortion bounds for functions in the family . We observe that the results so obtained for this main family can be viewed as extensions and generalizations for various subclasses of
and
.
2. Main Results
Theorem 2.1.
Let be such that
and
are given by (1.1). Then
if the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ14_HTML.gif)
is satisfied for some and
Proof.
In view of (1.5), we need to prove that , where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ15_HTML.gif)
Using the fact that , it sufficies to show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ16_HTML.gif)
Therefore, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ17_HTML.gif)
By hypothesis, last expression is nonnegative. Thus the proof is complete.
The coeficient bounds (2.1) is sharp for the function
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ18_HTML.gif)
where .
Corollary 2.2.
For ,
, if the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ19_HTML.gif)
holds, then .
Corollary 2.3.
For and
, if the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ20_HTML.gif)
holds, then .
Theorem 2.4.
Let be such that
and
are given by (1.13). Also, suppose that
and
. Then
(i)for ,
if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ21_HTML.gif)
(ii)for ,
if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ22_HTML.gif)
Proof.
According to Corollaries 2.2 and 2.3, we must show that if the condition (2.9) does not hold, then that is, we must have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ23_HTML.gif)
Choosing the values of on positive real axis where
and using
, the inequality (2.10) reduces to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ24_HTML.gif)
where denotes (k
+
)p[
−
+
],
denotes (k+
)p
−
+
and
denotes
Letting , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ25_HTML.gif)
If the condition (2.10) does not hold, then the numerator in (2.12) is negative for sufficiently close to 1. Hence there exists
in
for which (2.12) is negative.Therefore, it follows that
and so the proof is complete.
Theorem 2.5.
If , then for
,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ26_HTML.gif)
These bounds are sharp.
Proof.
Suppose . Let
and
. In view of (1.13), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ27_HTML.gif)
Using Theorem 2.4(i), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ28_HTML.gif)
The proofs of other cases are similar and so are omitted.
Corollary 2.6.
If , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ29_HTML.gif)
Theorem 2.7.
Suppose and
. Then
clco
if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ30_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ31_HTML.gif)
In particular, the extreme points of are
and
.
Proof.
Suppose . For functions of the form (2.17), we can write
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ32_HTML.gif)
On the other hand, for , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ33_HTML.gif)
Thus , by Theorem 2.4.
Conversely, suppose that . Then, it follows Theorem 2.4 that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ34_HTML.gif)
Setting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ35_HTML.gif)
and defining
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ36_HTML.gif)
where , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ37_HTML.gif)
Thus can be expressed as (2.17). The proof for the case
is similar and hence is omitted.
Theorem 2.8.
The class is closed under convex combinations.
Proof.
For let the functions
given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ38_HTML.gif)
are in . Also suppose the given fixed harmonic functions are given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ39_HTML.gif)
For the convex combinations of
can be expressed as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ40_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ41_HTML.gif)
(2.27) yields
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ42_HTML.gif)
Thus the coefficient estimate given by Theorem 2.4 holds. Therefore, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F759251/MediaObjects/13660_2009_Article_2005_Equ43_HTML.gif)
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Acknowledgment
This present investigation is supported with the Project no. DÜBAP-07-02-21 by Dicle University, The committee of the Scientific Research Projects.
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Ahuja, O.P., Güney, H.Ö. & Sakar, F.M. Certain Classes of Harmonic Multivalent Functions Based on Hadamard Product. J Inequal Appl 2009, 759251 (2009). https://doi.org/10.1155/2009/759251
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DOI: https://doi.org/10.1155/2009/759251