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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
for all and with
For , let
with
A function in is said to be p-valently starlike of order in that is, if and only if
for
Similarly, a function in is said to be p-valently convex of order in that is, if and only if
for
We denote by to be the family of functions in such that
for
Similarly, we denote by to be the family of functions in such that
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
for real number and
The class of -convex functions are defined by
We note that for and for .
Let
for real number and We note that
The class of functions is defined by as above:
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
if there exists a Schwarz function analytic in , with
and such that
In particular, if is univalent in , then
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
and suppose that either
(i) is starlike, or
(ii) is convex.
In addition, assume that
(iii)
If is analytic in with and
then and is the best dominant.
Lemma 2.2.
Let be univalent, and satisfies the following conditions:
for and for all For with in if
then and is the best dominant.
Proof.
Define the functions and by
in Lemma 2.1. Then, the functions
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
then
and is the best dominant.
Proof.
Let us put
where Then, we obtain easily the following result:
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:
for and for all For in if
then and is the best dominant.
Proof.
For real number, we define the functions and by
in Lemma 2.1. Then, the functions
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
Then,
and is the best dominant.
Proof.
Let us put
where Then, we have
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:
for and for all For in if
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
Then,
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
then and is the best dominant.
Proof.
In Corollary 2.6, we take
Corollary 2.9.
Let be convex. For if
Then,
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
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
then
and is the best dominant.
Proof.
In Theorem 2.3, we take
Corollary 2.12.
Let satisfies
where , then
and is the best dominant.
Proof.
In Theorem 2.5, we take
Corollary 2.13.
Let satisfies
where , then
and is the best dominant.
Proof.
In Corollary 2.12, we take
Corollary 2.14.
Let satisfies
where , then
and is the best dominant.
Proof.
In Corollary 2.13, we take
Corollary 2.15.
Let satisfies
where , then
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