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Sufficient Conditions for Subordination of Multivalent Functions

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

(1.1)

for all and with

For , let

(1.2)

with

A function in is said to be p-valently starlike of order in that is, if and only if

(1.3)

for

Similarly, a function in is said to be p-valently convex of order in that is, if and only if

(1.4)

for

We denote by to be the family of functions in such that

(1.5)

for

Similarly, we denote by to be the family of functions in such that

(1.6)

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

(1.7)

for real number and

The class of -convex functions are defined by

(1.8)

We note that for and for .

Let

(1.9)

for real number and We note that

The class of functions is defined by as above:

(1.10)

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

(1.11)

if there exists a Schwarz function analytic in , with

(1.12)

and such that

(1.13)

In particular, if is univalent in , then

(1.14)

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

(2.1)

and suppose that either

(i) is starlike, or

(ii) is convex.

In addition, assume that

(iii)

If is analytic in with and

(2.2)

then and is the best dominant.

Lemma 2.2.

Let be univalent, and satisfies the following conditions:

(2.3)

for and for all For with in if

(2.4)

then and is the best dominant.

Proof.

Define the functions and by

(2.5)

in Lemma 2.1. Then, the functions

(2.6)

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

(2.7)

then

(2.8)

and is the best dominant.

Proof.

Let us put

(2.9)

where Then, we obtain easily the following result:

(2.10)

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:

(2.11)

for and for all For in if

(2.12)

then and is the best dominant.

Proof.

For real number, we define the functions and by

(2.13)

in Lemma 2.1. Then, the functions

(2.14)

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

(2.15)

Then,

(2.16)

and is the best dominant.

Proof.

Let us put

(2.17)

where Then, we have

(2.18)

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:

(2.19)

for and for all For in if

(2.20)

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

(2.21)

Then,

(2.22)

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

(2.23)

then and is the best dominant.

Proof.

In Corollary 2.6, we take

Corollary 2.9.

Let be convex. For if

(2.24)

Then,

(2.25)

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

(2.26)

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

(2.27)

then

(2.28)

and is the best dominant.

Proof.

In Theorem 2.3, we take

Corollary 2.12.

Let satisfies

(2.29)

where , then

(2.30)

and is the best dominant.

Proof.

In Theorem 2.5, we take

(2.31)

Corollary 2.13.

Let satisfies

(2.32)

where , then

(2.33)

and is the best dominant.

Proof.

In Corollary 2.12, we take

Corollary 2.14.

Let satisfies

(2.34)

where , then

(2.35)

and is the best dominant.

Proof.

In Corollary 2.13, we take

Corollary 2.15.

Let satisfies

(2.36)

where , then

(2.37)

and is the best dominant.

Proof.

In Corollary 2.14, we take

References

  1. Yang D: On a criterion for multivalently starlikeness. Taiwanese Journal of Mathematics 1997,1(2):143–148.

    MATH  MathSciNet  Google Scholar 

  2. Özkan Ö, Altıntaş O: Applications of differential subordination. Applied Mathematics Letters 2006,19(8):728–734. 10.1016/j.aml.2005.09.002

    MATH  MathSciNet  Article  Google Scholar 

  3. 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 

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Correspondence to Öznur Özkan Kılıç.

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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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Keywords

  • Real Number
  • Analytic Function
  • Unit Disc
  • Multivalent Function
  • Schwarz Function