# Sufficient Conditions for Subordination of Multivalent Functions

- Öznur Özkan Kılıç
^{1}Email author

**2008**:374756

**DOI: **10.1155/2008/374756

© Öznur Özkan Kılıç. 2008

**Received: **13 January 2008

**Accepted: **28 March 2008

**Published: **21 April 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

*analytic in*, and let

for all and with

with

*p-valently starlike of order*in that is, if and only if

for

*p-valently convex of order*in that is, if and only if

for

for

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) .

for real number and

*convex*functions are defined by

We note that for and for .

for real number and We note that

*subordinate*to , written as

## 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]).

and suppose that either

(i) is starlike, or

(ii) is convex.

In addition, assume that

(iii)

then and is the best dominant.

Lemma 2.2.

then and is the best dominant.

Proof.

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.

and is the best dominant.

Proof.

Thus, using Lemma 2.1 and (2.7), we can obtain the result (2.8).

Lemma 2.4.

then and is the best dominant.

Proof.

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.

and is the best dominant.

Proof.

Thus, using (2.15) and Lemma 2.4, we can obtain the result (2.16).

Corollary 2.6.

then and is the best dominant.

Proof.

By putting in Lemma 2.4, we obtain Corollary 2.6.

Corollary 2.7.

and is the best dominant.

Proof.

By putting in Theorem 2.5, we obtain Corollary 2.7.

Corollary 2.8.

then and is the best dominant.

Proof.

In Corollary 2.6, we take

Corollary 2.9.

and is the best dominant.

Proof.

In Corollary 2.7, we take

Corollary 2.10.

then and is the best dominant.

Proof.

In Lemma 2.4, we take

Corollary 2.11.

and is the best dominant.

Proof.

In Theorem 2.3, we take

Corollary 2.12.

and is the best dominant.

Proof.

Corollary 2.13.

and is the best dominant.

Proof.

In Corollary 2.12, we take

Corollary 2.14.

and is the best dominant.

Proof.

In Corollary 2.13, we take

Corollary 2.15.

and is the best dominant.

Proof.

In Corollary 2.14, we take

## Authors’ Affiliations

## References

- Yang D:
**On a criterion for multivalently starlikeness.***Taiwanese Journal of Mathematics*1997,**1**(2):143–148.MATHMathSciNetGoogle Scholar - Özkan Ö, Altıntaş O:
**Applications of differential subordination.***Applied Mathematics Letters*2006,**19**(8):728–734. 10.1016/j.aml.2005.09.002MATHMathSciNetView ArticleGoogle Scholar - 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.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.