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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.2005.09.002Miller 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