﻿Some Properties of the Class of Univalent Functions with Negative Coefficients

Applied Mathematics
Vol.3 No.12(2012), Article ID:25341,6 pages DOI:10.4236/am.2012.312251

Some Properties of the Class of Univalent Functions with Negative Coefficients

Aisha Ahmed Amer, Maslina Darus*

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, Malaysia

Email: eamer_80@yahoo.com, *maslina@ukm.my

Received August 6, 2012; revised October 17, 2012; accepted October 26, 2012

Keywords: Analytic Function; Unit Disc; Coefficient Inequality; Closure Properties; Distortion Bound

ABSTRACT

The main object of this paper is to study some properties of certain subclass of analytic functions with negative coefficients defined by a linear operator in the open unit disc. These properties include the coefficient estimates, closure properties, distortion theorems and integral operators.

1. Introduction

Let be the class of analytic functions in the open unit disc

and be the subclass of consisting of functions of the form

Let denote the class of functions normalized by

(1)

which are analytic in the open unit disc. In particular,

For two functions given by (1) and given by

the Hadamard product (or convolution) is defined, as usual, by

Let the function be given by:

where denotes the Pochhammer symbol (or the shifted factorial) defined by:

Carlson and Shaffer [1] introduced a convolution operator on involving an incomplete beta function as:

(2)

Our work here motivated by Catas [2], who introduced an operator on as follows:

where

Now, using the Hadamard product (or convolution), the authors (cf. [3,4]) introduced the following linear operator:

Definition 1.1 Let

where

and is the Pochhammer symbol. We defines a linear operator by the following Hadamard product:

(3)

where

and the Pochhammer symbol .

Special cases of this operator include:

see [1].

• the Catas drivative operator [2]:

• the Ruscheweyh derivative operator [5] in the cases:

• the Salagean derivative operator [6]:

• the generalized Salagean derivative operator introduced by Al-Oboudi [7]:

• Note that:

Let denote the class of functions of the form

(4)

which are analytic in the open unit disc.

Following the earlier investigations by [8] and [9], we define -neighborhood of a function by

or,

where

Let denote the subclass of consisting of functions which satisfy

A function in is said to be starlike of order in.

A function is said to be convex of order it it satisfies

We denote by the subclass of consisting of all such functions [10].

The unification of the classes and is provided by the class of functions which also satisfy the following inequality

The class was investigated by Altintas [11].

Now, by using we will define a new class of starlike functions.

Definition 1.2 Let

A function belonging to is said to be in the class if and only if

(6)

Remark 1.3 The class is a generalization of the following subclasses:

i) and

defined and studied by [12];

ii) and studied by [13] and [14];

iii) studied by [15];

iv) studied by [16].

Now, we shall use the same method by [17] to establish certain coefficient estimates relating to the new introduced class.

2. Coefficient Estimates

Theorem 2.1 Let the function be defined by (1). Then belongs to the class if and only if

(7)

where

(8)

The result is sharp and the extremal functions are

(9)

Proof: Assume that the inequality (7) holds and let. Then we have

Consequently, by the maximum modulus theorem one obtains

Conversely,suppose that

.

Then from (6) we find that

Choose values of on the real axis such that

is real. Letting through real values, we obtain

or, equivalently

which gives (7).

Remark 2.2 In the special case Theorem 2.1 yields a result given earlier by [17].

Remark 2.3 In the special case Theorem 2.2 yields a result given earlier by [6].

Theorem 2.4 Let the function defined by (3) be in the class. Then

(10)

and

(11)

The equality in (10) and (11) is attained for the function given by (9).

Proof: By using Theorem 2.2, we find from (6) that

which immediately yields the first assertion (10) of Theorem 2.3.

On the other hand, taking into account the inequality (6), we also have

that is

which, in view of the coefficient inequality (10), can be put in the form

and this completes the proof of (11).

3. Closure Theorem

Theorem 3.1 Let the function be defined by

for be in the class then the function defined by

also belongs to the class, where

Proof: Since it follows from Theorem 2.1, that

Therefore,

Hence by Theorem 2.1, also.

Morever, we shall use the same method by [17] to prove the distrotion Theorems.

4. Distortion Theorems

Theorem 4.1  Let the function defined by (1) be in the class. Then we have

(12)

and

(13)

for, where and is given by (8).

The equalities in (12) and (13) are attained for the function given by

(14)

Proof: Note that if and only if

, where

By Theorem 2.2, we know that

that is

The assertions of (12) and (13) of Theorem 4.1 follow immediately. Finally, we note that the equalities (12) and (13) are attained for the function defined by

This completes the proof of Theorem 4.1.

Remark 4.2 In the special case Theorem 4.1 yields a result given earlier by [17].

Corollary 4.3  Let the function defined by (1) be in the class. Then we have

(15)

and

(16)

for. The equalities in (15) and (16) are attained for the function given in (14).

Corollary 4.4 Let the function defined by (1) be in the class. Then we have

(17)

and

(18)

for. The equalities in (17) and (18) are attained for the function given in (14).

Corollary 4.5 Let the function defined by (3) be in the class. Then the unit disc is mapped onto a domain that contains the disc

The result is sharp with the extremal function given in (14).

5. Integral Operators

Theorem 5.1 Let the function defined by (1) be in the class and let be a real number such that Then defined by

also belongs to the class

Proof: From the representation of it is obtained that

where

Therefore

since belongs to so by virtue of Theorem 2.1, is also element of

6. Acknowledgements

The work presented here was partially supported by LRGS/TD/2011/UKM/ICT/03/02.

REFERENCES

1. B. C. Carlson and D. B. Shaffer, “Starlike and Prestarlike Hypergeometric Functions,” SIAM Journal on Mathematical Analysis, Vol. 15, No. 4, 1984, pp. 737-745. doi:10.1137/0515057
2. A. Catas, “On a Certain Differential Sandwich Theorem Associated with a New Generalized Derivative Operator,” General Mathematics, Vol. 17, No. 4, 2009, pp. 83-95.
3. A. A. Amer and M. Darus, “A Distortion Theorem for a Certain Class of Bazilevic Function,” International Journal of Mathematical Analysis, Vol. 6, No. 12, 2012, pp. 591-597.
4. A. A. Amer and M. Darus, “On a Property of a Subclass of Bazilevic Functions,” Missouri Journal of Mathematical Sciences, In Press.
5. St. Ruscheweyh, “New Criteria for Univalent Functions,” Proceedings of the American Mathematical Society, Vol. 49, No. 1, 1975, pp. 109-115.
6. G. S. Salagean, “Subclasses of Univalent Functions,” Lecture Notes in Mathematics, Vol. 1013, 1983, pp. 362- 372. doi:10.1007/BFb0066543
7. F. M. Al-Oboudi, “On Univalent Functions Defined by a Generalized Salagean Operator,” International Journal of Mathematics and Mathematical Sciences, Vol. 2004, No. 27, 2004, pp. 1429-1436. doi:10.1155/S0161171204108090
8. A. W. Goodman, “Univalent Functions and Nonanalytic Curves,” Proceedings of the American Mathematical Society, Vol. 8, No. 3, 1957, pp. 598-601. doi:10.1090/S0002-9939-1957-0086879-9
9. St. Ruscheweyh, “Neighborhoods of Univalent Functions,” Proceedings of the American Mathematical Society, Vol. 81, No. 4, 1981, pp. 521-527. doi:10.1090/S0002-9939-1981-0601721-6
10. O. Alintas, H. Irmak and H. M. Srivastava, “Fractional Calculus and Certain Starlike Functions with Negative Coefficietns,” Computers & Mathematics with Applications, Vol. 30, No. 2, 1995, pp. 9-15. doi:10.1016/0898-1221(95)00073-8
11. O. Alintas, “On a Subclass of Certain Starlike Functions with Negative Coefficients,” Journal of the Mathematical Society of Japan, Vol. 36, 1991, pp. 489-495.
12. H. Silverman, “Univalent Functions with Negative Coefficients,” Proceedings of the American Mathematical Society, Vol. 51, No. 1, 1975, pp. 109-116. doi:10.1090/S0002-9939-1975-0369678-0
13. S. K. Chatterjea, “On Starlike Functions,” Journal of Pure Mathematics, Vol. 1, 1981, pp. 23-26.
14. H. M. Srivastava, S. Owa and S. K. Chatterjea, “A Note on Certain Classes of Starlike Functions,” Rendiconti del Seminario Matematico della Università di Padova, Vol. 77, 1987, pp.115-124.
15. M. D. Hur and G. H. Oh, “On Certain Class of Analytic Functions with Negative Coefficients,” Pusan Kyongnam Mathematical Journal, Vol. 5, No. 1, 1989, pp. 69-80.
16. M. Kamali, “Neighborhoods of a New Class of p-Valently Functions with Negative Coefficients,” Mathematical Inequalities & Applications, Vol. 9, No. 4, 2006, pp. 661- 670. doi:10.7153/mia-09-59
17. A. Catas, “Neighborhoods of a Certain Class of Analytic Functions with Negative Coefficients,” Banach Journal of Mathematical Analysis, Vol. 3, No. 1, 2009, pp. 111- 121.

NOTES

*Corresponding author.