Journal of Complex Analysis

Volume 2013 (2013), Article ID 403624, 6 pages

http://dx.doi.org/10.1155/2013/403624

## Subclasses of Harmonic Mappings Defined by Convolution

^{1}Department of Mathematics, Walchand College, Sangli, Maharashtra 416415, India^{2}Department of Mathematics, Willingdon College, Sangli, Maharashtra 416415, India

Received 29 November 2012; Accepted 27 February 2013

Academic Editor: Vladislav Kravchenko

Copyright © 2013 Santosh B. Joshi and Girish D. Shelake. 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

Two new subclasses of harmonic univalent functions defined by using convolution and integral convolution are introduced. These subclasses generate several known and new subclasses of harmonic univalent functions as special cases and provide a unified treatment in the study of these classes. Coefficient bounds, extreme points, distortion bounds, convolution conditions, and convex combination are also determined.

#### 1. Introduction

A continuous function is said to be a complex-valued harmonic function in a simply connected domain in complex plane if both real part of and imaginary part of are real harmonic in . Such functions can be expressed as where and are analytic in . We call the analytic part and the coanalytic part of . A necessary and sufficient condition for to be locally univalent and sense-preserving in is that for all in , see [1].

Every harmonic function is uniquely determined by the coefficients of power series expansions in the unit disk given by where for and for . For further information about these mappings, one may refer to [1–5].

In 1984, Clunie and Sheil-Small [1] studied the family of all univalent sense-preserving harmonic functions of the form (1) in , such that and are represented by (2). Note that reduces to the well-known family , the class of all normalized analytic univalent functions given in (2), whenever the coanalytic part of is zero. Let and denote the respective subclasses of and where the images of are convex. Denote by the subclass of for which .

The convolution of two functions of the form is given by and the integral convolution is defined by Towards the end of the last century, Jahangiri [3], Silverman [4], and Silverman and Silvia [5] were amongst those who focused on the harmonic starlike functions. Later Öztürk et. al. [6] defined the class consisting of functions such that and are of the forms which satisfy the condition for some and for all .

Several authors [3–16] have investigated various subclasses of harmonic functions. In this work, we introduce a new subclass of harmonic functions defined by convolution.

Let be a real constant with , then we denote , the subclass of of functions of the form that satisfy the condition where , , , , and are as given in (3).

We also denote , the subclass of of functions of the form that satisfy the condition where is real.

We note that the families and are of special interest, because they contain various classes of well-known harmonic univalent functions as well as many new ones. For different choice of , and we obtain the following various classes introduced by other authors:(1) (see Öztürk et. al [6]). (2) (see Jahangiri [3]). (3) (see Silverman and Silvia [5]). (4), with (see Avcı and Złotkiewicz [17] and Silverman [4]).(5) (see Jahangiri [3]). (6) (see Silverman [4]). (7) (see Dixit et al. [11]).(8) (see Ali et al. [7]). (9) (see Joshi et al. [14]).(10), where (see Jahangiri et al. [13]). (11), where (see Murugusundaramoorthy [15]). (12), where and (see Al-Shaqsi and Darus [10]). (13), where (see Murugusundaramoorthy et al. [16]).

It is clear that the class generates a number of known subclasses and provides a unified treatment of these subclasses of harmonic mappings. Motivated by work of Ali et al. [7], we obtain convolution characterization for functions in the class and . We also obtain sufficient coefficient condition for these two classes, and the last section is devoted to determine growth estimates and extreme points for the class and .

#### 2. Main Results

We now derive a convolution characterization for functions in the class .

Theorem 1. * Let . Then if and only if
*

*Proof. *A necessary and sufficient condition for to be in the class , with and of the form (1) is given by (8). The condition (8) holds if and only if

By simple algebraic manipulation, (11) gives
The latter condition, along with (8) for , establishes the result for all .

An application of the convolution condition in Theorem 1 gives sufficient condition for harmonic functions to belong to the class .

Theorem 2. *Let . Then if
*

*Proof. *For and given by (2), Theorem 1 gives
The last expression is nonnegative by hypothesis, and hence by Theorem 1, it follows that .

Theorem 3. *Let . Then if and only if
*

*Proof. *A necessary and sufficient condition for to be in the class , with and of the form (1) is given by (9). The condition (9) holds if and only if

By simple algebraic manipulation, we get the desired result.

Now sufficient coefficient condition for the class is easily obtained.

Theorem 4. *Let . Then if
*

We further let and denote the subclasses of and , respectively, consisting of functions such that and are of the form Let with .

Theorem 5. *Let of the form (18). Then if and only if
*

*Proof. *If , then (8) is equivalent to
for . Letting through real values, we get
which gives required condition (20).

Conversely, for and given by (18),
which is nonnegative by hypothesis.

Theorem 6. *Let of the form (18). Then if and only if
*

The following theorem gives the distortion bounds for functions in , and which gives a covering result for the classes and , respectively.

Theorem 7. *Let and , then for , we have
**
The result is sharp with equality for .*

*Proof. *We have
Thus,
The proof of left-hand inequality follows in lines similar to that of right-hand side inequality.

Theorem 8. *Let and , then for , we have
**
The result is sharp with equality for .*

Corollary 9. *Let , then
*

Corollary 10. *Let , then
*

Finally, we determine the extreme points of the class , and .

Theorem 11. *Let
**
A function if and only if can be expressed in the form
**
where , , and . In particular, the extreme points of are and .*

*Proof. *Let
Since
from Theorem 5, .

Conversely, if , then and . Set , , , and .

Then,

Theorem 12. *Let
**
A function if and only if can be expressed in the form
**
where , , , and . In particular, the extreme points of are and .*

#### Acknowledgment

The authors are thankful to the referees for their valuable suggestions and comments.

#### References

- J. Clunie and T. Sheil-Small, “Harmonic univalent functions,”
*Annales Academiae Scientiarum Fennicae A*, vol. 9, pp. 3–25, 1984. View at Google Scholar · View at MathSciNet - P. Duren,
*Harmonic Mappings in the Plane*, vol. 156 of*Cambridge Tracts in Mathematics*, Cambridge University Press, Cambridge, UK, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - J. M. Jahangiri, “Harmonic functions starlike in the unit disk,”
*Journal of Mathematical Analysis and Applications*, vol. 235, no. 2, pp. 470–477, 1999. View at Publisher · View at Google Scholar · View at MathSciNet - H. Silverman, “Harmonic univalent functions with negative coefficients,”
*Journal of Mathematical Analysis and Applications*, vol. 220, no. 1, pp. 283–289, 1998. View at Publisher · View at Google Scholar · View at MathSciNet - H. Silverman and E. M. Silvia, “Subclasses of harmonic univalent functions,”
*New Zealand Journal of Mathematics*, vol. 28, no. 2, pp. 275–284, 1999. View at Google Scholar · View at MathSciNet - M. Öztürk, S. Yalçin, and M. Yamankaradeniz, “Convex subclass of harmonic starlike functions,”
*Applied Mathematics and Computation*, vol. 154, no. 2, pp. 449–459, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - R. M. Ali, B. A. Stephen, and K. G. Subramanian, “Subclasses of harmonic mappings defined by convolution,”
*Applied Mathematics Letters*, vol. 23, no. 10, pp. 1243–1247, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - R. M. Ali, M. H. Khan, V. Ravichandran, and K. G. Subramanian, “A class of multivalent functions with negative coefficients defined by convolution,”
*Bulletin of the Korean Mathematical Society*, vol. 43, no. 1, pp. 179–188, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - O. P. Ahuja, “Planar harmonic univalent and related mappings,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 6, no. 4, article 122, 18 pages, 2005. View at Google Scholar · View at MathSciNet - K. Al-Shaqsi and M. Darus, “On harmonic functions defined by derivative operator,”
*Journal of Inequalities and Applications*, vol. 2008, Article ID 263413, 10 pages, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - K. K. Dixit, A. L. Pathak, S. Porwal, and R. Agarwal, “On a subclass of harmonic univalent functions defined by convolution and integral convolution,”
*International Journal of Pure and Applied Mathematics*, vol. 69, no. 3, pp. 255–264, 2011. View at Google Scholar · View at MathSciNet - B. A. Frasin, “Comprehensive family of harmonic univalent functions,”
*SUT Journal of Mathematics*, vol. 42, no. 1, pp. 145–155, 2006. View at Google Scholar · View at MathSciNet - J. M. Jahangiri, G. Murugusundaramoorthy, and K. Vijaya, “Salagean-type harmonic univalent functions,”
*Southwest Journal of Pure and Applied Mathematics*, vol. 2002, no. 2, pp. 77–82, 2002. View at Google Scholar · View at MathSciNet - S. B. Joshi, G. D. Shelake, and A. H. Suzeini, “On a subclass of harmonic univalent functions defined by convolution,” communicated.
- G. Murugusundaramoorthy, “A class of Ruscheweyh-type harmonic univalent functions with varying arguments,”
*Southwest Journal of Pure and Applied Mathematics*, no. 2, pp. 90–95, 2003. View at Google Scholar · View at MathSciNet - G. Murugusundaramoorthy, K. Vijaya, and B. A. Frasin, “A subclass of harmonic functions with negative coefficients defined by Dziok-Srivastava operator,”
*Tamkang Journal of Mathematics*, vol. 42, no. 4, pp. 463–473, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Avcı and E. Złotkiewicz, “On harmonic univalent mappings,”
*Annales Universitatis Mariae Curie-Skłodowska*, vol. 44, pp. 1–7, 1990. View at Google Scholar · View at MathSciNet