International Journal of Mathematics and Mathematical Sciences

Volume 2014 (2014), Article ID 358467, 5 pages

http://dx.doi.org/10.1155/2014/358467

Research Article

Some Properties of Certain Class of Analytic Functions

Uzoamaka A. Ezeafulukwe1,2 and Maslina Darus²

¹Mathematics Department, Faculty of Physical Sciences, University of Nigeria, Nsukka, Nigeria

²School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

Received 21 April 2014; Revised 18 June 2014; Accepted 27 June 2014; Published 10 July 2014

Academic Editor: Hari M. Srivastava

Copyright © 2014 Uzoamaka A. Ezeafulukwe and Maslina Darus. 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.

Abstract.

We obtain some properties related to the coefficient bounds for certain subclass of analytic functions. We also work on the differential subordination for a certain class of functions.

1. Introductions

Let denote the class of functions

which is analytic in the unit disc . Let

Now let be the class of functions defined by

The Hadamard product of two functions and is defined by

where and are analytic in .

Let , , and then is analytic in the open unit disc . The function defined in (3) is equivalent to

where is the Hadamard product and is analytic in the open unit disc .

We introduce a class of functions

where

Authors like Saitoh [1] and Owa [2, 3] had previously studied the properties of the class of functions . They obtained many interesting results and Wang et al. [4] studied the extreme points, coefficient bounds, and radius of univalency of the same class of functions. They obtained the following theorem among other results.

Theorem 1 (see [4]). Let . A function if and only if can be expressed as

where is the probability measure defined on

For fixed , , and , the class and the probability measure defined on are one-to-one by expression (8).

Recently, Hayami et al. [5] studied the coefficient estimates of the class of function in the open unit disc . They derived results based on properties of the class of functions , . Xu et al. [6] used the principle of differential subordination and the Dziok-Srivastava convolution operator to investigate some analytic properties of certain subclass of analytic functions. We also note that Stanciu et al. [7] used the properties of the class of functions , , to investigate the analytic and univalent properties of the following integral operator:

where .

Motivated by the work in [1–7], we used the properties of the class of function , , to investigate the coefficient estimates of the class of functions in the open unit disc . We also use the principle of differential subordination to investigate some properties of the class of functions .

We state the following known results required to prove our work.

Definition 2. If and are analytic in , then is said to be subordinate to , written as or . If is univalent in , then and .

Theorem 3 (see [8]). Consider if and only if there is probability measure on such that

and . The correspondence between and the set of probability measures on given by Hallenbeck [9] is one-to-one.

Theorem 4 (see [10, 11]). Let be convex in , , , and . If and

then

The function is convex and the best -dominant.

Lemma 5 (see [10]). Let be starlike in , with and . If satisfies

then

and is the best -dominant.

Lemma 6 (see [12]). Let , with in . Then, for ,

(i),(ii).

Remark 7. The combination (i) and (ii) of Lemma 6 gives

Remark 8. For convenience, we limit our result to the principal branch and otherwise stated the constrains on , , , , , and which remain the same throughout this paper.

2. Coefficient Bounds of the Class of Functions

We begin with the following result.

Theorem 9. Let be as defined in (3). A function , if and only if can be expressed as

where and is the probability measure defined on .

Proof. If , then

By Theorem 3,

and (19) can be written as

which yields

and so the expression (17).

If can be expressed as (17), reverse calculation shows that .

Corollary 10. Let be defined as in (3). A function if and only if can be expressed as

where is the probability measure defined on .

Proof. It is as in Theorem 9.

Corollary 11. Let be as defined in (3). If , then, for and , we have

where

Proof. Let from (17) and

Comparing the coefficient yields the result.

Theorem 12. Let and . Then for we have

Proof. Since , then

and then

where

From (23) and (28), we got

The application of Remark 7 to (27) gives

Since

then Theorem 12 is proved.

3. Application of Differential Subordination to the Function

Here we calculate some subordinate properties of the class .

Theorem 13. Let and let be starlike in with and . If

then

and is the best -dominant.

Proof. Let ; then

Since is analytic in and , it suffices to show that

Following the same argument in [10] (pages 76 and 77), (36) is true. Application of Lemma 5 proves Theorem 13 with as the best -dominant.

Example 14. Let ; if

then

and is the best -dominant.

Solution. If and , , then simple calculation shows that and is starlike and the argument in [10] shows (37). The proof also follows from Lemma 5.

Theorem 15. Let and , with . If

then

Proof. Let , and, from (6),

where

Let ; then , where , and from (39) we have

and is convex and univalent in . So, by Lemma 6,

This completes the proof of Theorem 15.

Corollary 16. Let

If

then

Proof. The result follows from Theorem 15.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

Both authors read and approved the final paper.

Acknowledgments

The work here is fully supported by LRGS/TD/2011/UKM/ICT/03/02 and GUP-2013-004. The authors also would like to thank the referee and editor in charge for the comments and suggestions given to improve their paper.

References

H. Saitoh, “On inequalities for certain analytic functions,” Mathematica Japonica, vol. 35, no. 6, pp. 1073–1076, 1990.

S. Owa, “Some properties of certain analytic functions,” Soochow Journal of Mathematics, vol. 13, no. 2, pp. 197–201, 1987.

S. Owa, “Generalization properties for certain analytic functions,” International Journal of Mathematics and Mathematical Sciences, vol. 21, no. 4, pp. 707–712, 1998.

Z. Wang, C. Gao, and S. Yuan, “On the univalency of certain analytic functions,” Journal of Inequalities in Pure and Applied Mathematics, vol. 7, no. 1, pp. 1–4, 2006.

H. Hayami, S. Owa, and H. M. Srivastava, “Coefficient estimates for a certain class of analytic functions involving the arguments of their derivative,” Jnanabha, vol. 43, pp. 37–43, 2013.

Q.-H. Xu, H.-G. Xiao, and H. M. Srivastava, “Some applications of differential sub-ordination and the Dziok-Srivastava convolution operator,” Applied Mathematics and Computation, vol. 230, pp. 496–508, 2014.

L. F. Stanciu, D. Breaz, and H. M. Srivastava, “Some criteria for univalence of a certain integral operator,” Novi Sad Journal of Mathematics, vol. 43, pp. 51–57, 2013.

D. J. Hallenbeck and T. H. MacGregor, Linear Problems and Convexity Techniques in Geometric Function Theory, vol. 22, Pitman, Boston, Mass, USA, 1984.

D. J. Hallenbeck, “Convex hulls and extreme points of some families of univalent functions,” Transactions of the American Mathematical Society, vol. 192, pp. 285–292, 1974.

S. S. Miller and P. T. Mocanu, Differential Subordinations: Theory and Applications, vol. 225 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2000.

S. S. Miller and P. T. Mocanu, “Differential subordinations and univalent functions,” The Michigan Mathematical Journal, vol. 28, no. 2, pp. 157–172, 1981.

A. W. Goodman, Univalent Functions, vol. 1, Polygon Publishing House, Washington, DC, USA, 1983.