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Sufficient Conditions for Subordination of Multivalent Functions
Journal of Inequalities and Applications volume 2008, Article number: 374756 (2008)
Abstract
The authors investigate various subordination results for some subclasses of analytic functions in the unit disc. We obtain some sufficient conditions for multivalent close-to-starlikeness.
1. Introduction and Definitions
Let and let
be the set of all functions analytic in
, and let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ1_HTML.gif)
for all and
with
For , let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ2_HTML.gif)
with
A function in
is said to be p-valently starlike of order
in
that is,
if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ3_HTML.gif)
for
Similarly, a function in
is said to be p-valently convex of order
in
that is,
if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ4_HTML.gif)
for
We denote by to be the family of functions
in
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ5_HTML.gif)
for
Similarly, we denote by to be the family of functions
in
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ6_HTML.gif)
for
We note that the classes and
are special classes of the class of p-valently close-to-convex of order
, the class of p-valently close-to-starlike of order
in
, respectively.
In particular, the classes ,
,
,
,
are the familiar classes of univalent, starlike, convex, close-to-convex, and close-to-starlike functions in
respectively. Also, we note that
(i);
(ii).
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ7_HTML.gif)
for real number and
The class of -convex functions are defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ8_HTML.gif)
We note that for
and
for
.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ9_HTML.gif)
for real number and
We note that
The class of functions is defined by as above:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ10_HTML.gif)
A class defined by was studied by Dinggong [1], and also, for
, the general case of
was studied by Özkan and Altıntaş [2]. Given two functions
and
, which are analytic in
, the function
is said to be subordinate to
, written as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ11_HTML.gif)
if there exists a Schwarz function analytic in
, with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ12_HTML.gif)
and such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ13_HTML.gif)
In particular, if is univalent in
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ14_HTML.gif)
2. The Main Results
In proving our main results, we need the following lemma due to Miller and Mocanu.
Lemma 2.1 (see [3, page 132]).
Let be univalent in
and let
and
be analytic in a domain
containing
with
when
Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ15_HTML.gif)
and suppose that either
(i) is starlike, or
(ii) is convex.
In addition, assume that
(iii)
If is analytic in
with
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ16_HTML.gif)
then and
is the best dominant.
Lemma 2.2.
Let be univalent,
and satisfies the following conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ17_HTML.gif)
for and for all
For
with
in
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ18_HTML.gif)
then and
is the best dominant.
Proof.
Define the functions and
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ19_HTML.gif)
in Lemma 2.1. Then, the functions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ20_HTML.gif)
Using (2.3), we obtain that is starlike in
and
for all
Since it satisfies preconditions of Lemma 2.1 and using (2.4), it follows from Lemma 2.1 that
and
is the best dominant.
Theorem 2.3.
Let be univalent,
and satisfies the conditions (2.3) in Lemma 2.2. For
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ21_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ22_HTML.gif)
and is the best dominant.
Proof.
Let us put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ23_HTML.gif)
where Then, we obtain easily the following result:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ24_HTML.gif)
Thus, using Lemma 2.1 and (2.7), we can obtain the result (2.8).
Lemma 2.4.
Let be univalent and satisfies the following conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ25_HTML.gif)
for and for all
For
in
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ26_HTML.gif)
then and
is the best dominant.
Proof.
For real number, we define the functions
and
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ27_HTML.gif)
in Lemma 2.1. Then, the functions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ28_HTML.gif)
Using (2.11), we obtain that is starlike in
and
for all
Since it satisfies preconditions of Lemma 2.1 and using (2.12), it follows from Lemma 2.1 that
and
is the best dominant.
Theorem 2.5.
Let be univalent and satisfies the conditions (2.11) in Lemma 2.4. For
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ29_HTML.gif)
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ30_HTML.gif)
and is the best dominant.
Proof.
Let us put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ31_HTML.gif)
where Then, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ32_HTML.gif)
Thus, using (2.15) and Lemma 2.4, we can obtain the result (2.16).
Corollary 2.6.
Let be univalent and satisfies the following conditions:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ33_HTML.gif)
for and for all
For
in
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ34_HTML.gif)
then and
is the best dominant.
Proof.
By putting in Lemma 2.4, we obtain Corollary 2.6.
Corollary 2.7.
Suppose satisfies the conditions (2.19) in Corollary 2.6. For
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ35_HTML.gif)
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ36_HTML.gif)
and is the best dominant.
Proof.
By putting in Theorem 2.5, we obtain Corollary 2.7.
Corollary 2.8.
Let be univalent;
is convex for all
For
in
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ37_HTML.gif)
then and
is the best dominant.
Proof.
In Corollary 2.6, we take
Corollary 2.9.
Let be convex. For
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ38_HTML.gif)
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ39_HTML.gif)
and is the best dominant.
Proof.
In Corollary 2.7, we take
Corollary 2.10.
Let be univalent,
is convex for all
For
in
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ40_HTML.gif)
then and
is the best dominant.
Proof.
In Lemma 2.4, we take
Corollary 2.11.
Let be univalent,
is convex, for all
If
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ41_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ42_HTML.gif)
and is the best dominant.
Proof.
In Theorem 2.3, we take
Corollary 2.12.
Let satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ43_HTML.gif)
where , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ44_HTML.gif)
and is the best dominant.
Proof.
In Theorem 2.5, we take
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ45_HTML.gif)
Corollary 2.13.
Let satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ46_HTML.gif)
where , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ47_HTML.gif)
and is the best dominant.
Proof.
In Corollary 2.12, we take
Corollary 2.14.
Let satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ48_HTML.gif)
where , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ49_HTML.gif)
and is the best dominant.
Proof.
In Corollary 2.13, we take
Corollary 2.15.
Let satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ50_HTML.gif)
where , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F374756/MediaObjects/13660_2008_Article_1805_Equ51_HTML.gif)
and is the best dominant.
Proof.
In Corollary 2.14, we take
References
Yang D: On a criterion for multivalently starlikeness. Taiwanese Journal of Mathematics 1997,1(2):143–148.
Özkan Ö, Altıntaş O: Applications of differential subordination. Applied Mathematics Letters 2006,19(8):728–734. 10.1016/j.aml.2005.09.002
Miller SS, Mocanu PT: Differential Subordinations: Theory and Applications, Monographs and Textbooks in Pure and Applied Mathematics. Volume 225. Marcel Dekker, New York, NY, USA; 2000:xii+459.
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Özkan Kılıç, Ö. Sufficient Conditions for Subordination of Multivalent Functions. J Inequal Appl 2008, 374756 (2008). https://doi.org/10.1155/2008/374756
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DOI: https://doi.org/10.1155/2008/374756