Open Access

Subclasses of Meromorphic Functions Associated with Convolution

  • Maisarah Haji Mohd1,
  • Rosihan M. Ali1,
  • Lee See Keong1Email author and
  • V. Ravichandran2
Journal of Inequalities and Applications20092009:190291

https://doi.org/10.1155/2009/190291

Received: 12 December 2008

Accepted: 11 March 2009

Published: 16 March 2009

Abstract

Several subclasses of meromorphic functions in the unit disk are introduced by means of convolution with a given fixed meromorphic function. Subjecting each convoluted-derived function in the class to be subordinated to a given normalized convex function with positive real part, these subclasses extend the classical subclasses of meromorphic starlikeness, convexity, close-to-convexity, and quasi-convexity. Class relations, as well as inclusion and convolution properties of these subclasses, are investigated.

1. Introduction

Let be the set of all analytic functions defined in the unit disk . We denote by the class of normalized analytic functions defined in . For two functions and analytic in , the function is subordinate to , written as
(1.1)

if there exists a Schwarz function , analytic in with and such that In particular, if the function is univalent in , then is equivalent to and

A function is starlike if is subordinate to and convex if is subordinate to . Ma and Minda [1] gave a unified presentation of these classes and introduced the classes
(1.2)

where is an analytic function with positive real part, , and maps the unit disk onto a region starlike with respect to 1.

The convolution or the Hadamard product of two analytic functions and is given by
(1.3)

In term of convolution, a function is starlike if is starlike, and convex if is starlike. These ideas led to the study of the class of all functions such that is starlike for some fixed function in In this direction, Shanmugam [2] introduced and investigated various subclasses of analytic functions by using the convex hull method [35] and the method of differential subordination. Ravichandran [6] introduced certain classes of analytic functions with respect to -ply symmetric points, conjugate points, and symmetric conjugate points, and also discussed their convolution properties. Some other related studies were also made in [79], and more recently by Shamani et al. [10].

Let denote the class of meromorphic functions of the form
(1.4)
which are analytic and univalent in the punctured unit disk . For we recall that the classes of meromorphic starlike, meromorphic convex, meromorphic close-to-convex, meromorphic -convex (Mocanu sense), and meromorphic quasi-convex functions of order , denoted by , , , and respectively, are defined by
(1.5)
The convolution of two meromorphic functions and , where is given by (1.4) and , is given by
(1.6)

Motivated by the investigation of Shanmugam [2], Ravichandran [6], and Ali et al. [7, 11], several subclasses of meromorphic functions defined by means of convolution with a given fixed meromorphic function are introduced in Section 2. These new subclasses extend the classical classes of meromorphic starlike, convex, close-to-convex, -convex, and quasi-convex functions given in (1.5). Section 3 is devoted to the investigation of the class relations as well as inclusion and convolution properties of these newly defined classes.

We will need the following definition and results to prove our main results.

Let denote the class of starlike functions of order . The class of prestarlike functions of order is defined by
(1.7)
for , and
(1.8)

Theorem 1.1 (see [12, Theorem 2.4]).

Let , , and . Then, for any analytic function ,
(1.9)
,

where denotes the closed convex hull of .

Theorem 1.2 (see [13]).

Let be convex in and with . If is analytic in with , then
(1.10)

2. Definitions

In this section, various subclasses of are defined by means of convolution and subordination. Let be a fixed function in , and let be a convex univalent function with positive real part in and .

Definition 2.1.

The class consists of functions satisfying in and the subordination
(2.1)

Remark 2.2.

If , then coincides with , where
(2.2)

Definition 2.3.

The class consists of functions satisfying in and the subordination

(2.3)

Definition 2.4.

The class consists of functions such that in for some and satisfying the subordination

(2.4)

Definition 2.5.

For real, the class consists of functions satisfying , in and the subordination

(2.5)

Definition 2.6.

The class consists of functions such that in for some and satisfying the subordination

(2.6)

3. Main Results

This section is devoted to the investigation of class relations as well as inclusion and convolution properties of the new subclasses given in Section 2.

Theorem 3.1.

Let be a convex univalent function satisfying , and with . If , then . Equivalently, if , then .

Proof.

Define the function by
(3.1)
For , it follows that
(3.2)
and therefore,
(3.3)
Hence . A computation shows that
(3.4)
Theorem 1.1 yields
(3.5)
and because , it follows that
(3.6)

Theorem 3.2.

The function if and only if

Proof.

The results follow from the equivalence relations
(3.7)

Theorem 3.3.

Let be a convex univalent function satisfying and with . If , then .

Proof.

Since , it follows that
(3.8)
and thus
(3.9)
Let
(3.10)
A similar computation as in the proof of Theorem 3.1 yields
(3.11)
Inequality (3.9) shows that . Therefore Theorem 1.1 yields
(3.12)

hence .

Corollary 3.4.

under the conditions of Theorem 3.3 .

Proof.

The proof follows from (3.12).

In particular, when , the following corollary is obtained.

Corollary 3.5.

Let and satisfy the conditions of Theorem 3.3 . If , then .

Theorem 3.6.

Let and satisfy the conditions of Theorem 3.3. If , then . Equivalently .

Proof.

If , it follows from Theorem 3.2 that . Theorem 3.3 shows that Hence .

Theorem 3.7.

Under the conditions of Theorem 3.3, if with respect to , then with respect to .

Proof.

Theorem 3.3 shows that . Since , (3.9) yields .

Let the function be defined by
(3.13)
A similar computation as in the proof of Theorem 3.1 yields
(3.14)
Since and , it follows from Theorem 1.1 that
(3.15)

Thus with respect to .

Corollary 3.8.

under the assumptions of Theorem 3.3 .

Proof.

The subordination (3.15) shows that .

Theorem 3.9.

Let . Then

(i) ,

(ii) for .

Proof.

Define the function by
(3.16)
and the function by
(3.17)
For , it follows that Note also that
(3.18)
  1. (i)
    Since and
    (3.19)
     
Theorem 1.2 yields Hence .
  1. (ii)
    Observe that
    (3.20)
     

Furthermore and from (i). Since and is convex, we deduce that . Therefore,

Corollary 3.10.

The class is a subset of the class

Proof.

The proof follows from the definition of the classes by taking .

Theorem 3.11.

The function if and only if .

Proof.

If , then there exists such that
(3.21)
Also,
(3.22)

Since , by Theorem 3.2 , . Hence .

Conversely, if , then
(3.23)
for some . Let be such that . The proof is completed by observing that
(3.24)

Corollary 3.12.

Let and satisfy the conditions of Theorem 3.3 . If , then

Proof.

If , Theorem 3.11 gives Theorem 3.7 next gives Thus, Theorem 3.11 yields

Corollary 3.13.

under the conditions of Theorem 3.3.

Proof.

If , it follows from Corollary 3.12 that The subordination

(3.25)

gives . Therefore

Open Problem

An analytic convex function in the unit disk is necessarily starlike. For the meromorphic case, is it true that ?

Declarations

Acknowledgment

The work presented here was supported in part by the USM's Reserach University grant and the FRGS grant

Authors’ Affiliations

(1)
School of Mathematical Sciences, Universiti Sains Malaysia
(2)
Department of Mathematics, University of Delhi

References

  1. Ma WC, Minda D: A unified treatment of some special classes of univalent functions. In Proceedings of the Conference on Complex Analysis (Tianjin, 1992), 1994, Cambridge, Mass, USA, Conference Proceedings and Lecture Notes in Analysis, I. International Press; 157–169.MATHGoogle Scholar
  2. Shanmugam TN: Convolution and differential subordination. International Journal of Mathematics and Mathematical Sciences 1989,12(2):333–340. 10.1155/S0161171289000384MathSciNetView ArticleMATHGoogle Scholar
  3. Barnard RW, Kellogg C: Applications of convolution operators to problems in univalent function theory. The Michigan Mathematical Journal 1980,27(1):81–94.MathSciNetView ArticleMATHGoogle Scholar
  4. Parvatham R, Radha S: On -starlike and -close-to-convex functions with respect to -symmetric points. Indian Journal of Pure and Applied Mathematics 1986,17(9):1114–1122.MathSciNetMATHGoogle Scholar
  5. Ruscheweyh S, Sheil-Small T: Hadamard products of Schlicht functions and the Pólya-Schoenberg conjecture. Commentarii Mathematici Helvetici 1973, 48: 119–135. 10.1007/BF02566116MathSciNetView ArticleMATHGoogle Scholar
  6. Ravichandran V: Functions starlike with respect to -ply symmetric, conjugate and symmetric conjugate points. The Journal of the Indian Academy of Mathematics 2004,26(1):35–45.MathSciNetMATHGoogle Scholar
  7. Ali RM, Ravichandran V, Lee SK: Subclasses of multivalent starlike and convex functions. to appear in Bulletin of the Belgian Mathematical Society. Simon Stevin to appear in Bulletin of the Belgian Mathematical Society. Simon StevinGoogle Scholar
  8. Padmanabhan KS, Parvatham R: Some applications of differential subordination. Bulletin of the Australian Mathematical Society 1985,32(3):321–330. 10.1017/S0004972700002410MathSciNetView ArticleMATHGoogle Scholar
  9. Padmanabhan KS, Parvatham R: Some applications of differential subordination. Bulletin of the Australian Mathematical Society 1985,32(3):321–330. 10.1017/S0004972700002410MathSciNetView ArticleMATHGoogle Scholar
  10. Shamani S, Ali RM, Ravichandran V, Lee SK: Convolution and differential subordination for multivalent functions. to appear in Bulletin of the Malaysian Mathematical Sciences Society. Second Series to appear in Bulletin of the Malaysian Mathematical Sciences Society. Second SeriesGoogle Scholar
  11. Ali RM, Ravichandran V, Seenivasagan N: Subordination and superordination of the Liu-Srivastava linear operator on meromorphic functions. Bulletin of the Malaysian Mathematical Sciences Society. Second Series 2008,31(2):193–207.MathSciNetMATHGoogle Scholar
  12. Ruscheweyh S: Convolutions in Geometric Function Theory: Fundamental Theories of Physics, Séminaire de Mathématiques Supérieures. Volume 83. Presses de l'Université de Montréal, Montreal, Canada; 1982:168.Google Scholar
  13. Eenigenburg P, Miller SS, Mocanu PT, Reade MO: On a Briot-Bouquet differential subordination. Revue Roumaine de Mathématiques Pures et Appliquées 1984,29(7):567–573.MathSciNetMATHGoogle Scholar

Copyright

© Maisarah Haji Mohd et al. 2009

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