/ / Article
Special Issue

## Analytic and Harmonic Univalent Functions

View this Special Issue

Research Article | Open Access

Volume 2014 |Article ID 467929 | 7 pages | https://doi.org/10.1155/2014/467929

# On Certain Subclass of Harmonic Starlike Functions

Accepted19 Mar 2014
Published09 Apr 2014

#### Abstract

Coefficient conditions, distortion bounds, extreme points, convolution, convex combinations, and neighborhoods for a new class of harmonic univalent functions in the open unit disc are investigated. Further, a class preserving integral operator and connections with various previously known results are briefly discussed.

#### 1. Introduction

A continuous complex-valued function is said to be harmonic in a simply connected domain if both and are real harmonic in . There is a close interrelation between analytic functions and harmonic functions. For example, for real harmonic functions and , there exist analytic functions and so that and . Then , where and are, respectively, the analytic functions and . In this case, the Jacobian of is given by . The mapping is orientation preserving and locally one-to-one in if and only if in . The function is said to be harmonic univalent in if the mapping is orientation preserving, harmonic, and one-to-one in . We call the analytic part and the coanalytic part of (see Clunie and Sheil-Small ).

Denote by the class of functions that are harmonic univalent and orientation preserving in the open unit disk for which . Then for , we may express the analytic functions and as Note that reduces to the class of normalized analytic univalent functions if the coanalytic part of its members is zero. For this class the function may be expressed as A function with and given by (1) is said to be harmonic starlike of order for , if The class of all harmonic starlike functions of order is denoted by and extensively studied by Jahangiri . The cases and were studied by Silverman and Silvia  and Silverman . Other related works of the class also appeared in .

Definition 1. Let where and are given by (1). Let and . Then if and only if

We note that for , the class reduces to the class . Further, if the coanalytic part is zero, the class reduces to the class of functions which satisfy the condition for some , , and . Observe that the classes and were introduced and studied by many authors and these include, for example, by Obradovic and Joshi , Padmanabhan , Li and Owa , Xu and Yang , Singh and Gupta , and Lashin . We also note that for , the class was studied by Silverman .

#### 2. Main Results

The first theorem of this section determines the sufficient coefficient condition for functions to belong to the class . The following lemma obtained by Jahangiri is needed.

Lemma 2 (see [2, Theorem 1]). Let with and of the form (1) and let where . Then is harmonic, orientation preserving, and univalent in , and .

Theorem 3. Let where and are of the form (1). If for some , and , then is harmonic, orientation preserving, and univalent in and .

Proof. Since and , , it follows from Lemma 2 that and hence is harmonic, orientation preserving, and univalent in . Now, we only need to show that if (7) holds then Using the fact that if and only if , it suffices to show that where Substituting for and in (9), we obtain by the given condition (7).
The harmonic function where shows that the coefficient bound given in (7) is sharp. The functions of the form (12) are in since

Remark 4. Setting in Theorem 3 yields the result obtained by Lashin [22, Theorem 2.1].

We denote by the class of functions whose coefficients satisfy the condition (7).

Theorem 5. Let and . Then .

Proof. For , it follows from (7) that Hence .

As a consequence of Theorem 5, the functions in are starlike harmonic in .

Corollary 6. For and ,.

#### 3. Distortion Bounds and Extreme Points

In this section, we obtain the distortion bounds and extreme points for functions in the class .

Theorem 7. Let where and are of the form (1) and . Then for , we have where The result is sharp.

Proof. We shall prove the first inequality. Let . Then we have and so
The proof of the inequality (16) is similar to the proof of the inequality (15), thus we omit it.
The upper bound given for is sharp and the equality occurs for the function where . This completes the proof of Theorem 7.

Now, we determine the extreme points of the closed convex hull of the class denoted by .

Theorem 8. Let where and are given by (1). Then if and only if where In particular, the extreme points of the class are and , respectively.

Proof. For a function of the form (21), we have But Thus .
Conversely, suppose that . Set Then by the inequality (7), we have and . Define and note that . Thus we obtain . This completes the proof of the theorem.

#### 4. Convolution and Convex Combinations

For two harmonic functions we define their convolution Using this definition, we show that the class is closed under convolution.

Theorem 9. For and , let . Then .

Proof. We note that and . For the convolution , we have Therefore .

We show that the class is closed under convex combination of its members.

Theorem 10. The class is closed under convex combination.

Proof. For , let where is given by Then by (7), we have For , , the convex combination of may be written as Then by (7), we have Therefore .

#### 5. Neighborhood Results

Following the earlier investigations by Goodman , Ruscheweyh , Altintas et al. , and Porwal and Aouf , we define the -neighborhood of function by In particular, for the identity function , we immediately have

Theorem 11. , where

Proof. Let . Then, in view of (7), since and are increasing functions of , we have which yields On the other hand, we also find from (7) From (37) and (38), we obtain which is equivalent to

#### 6. A Family of Class Preserving Integral Operator

In this section, we consider the closure property of the class under the Bernardi integral operator , which is defined by

Theorem 12. Let be in the class , where and are given by (1). Then defined by (41) also belongs to the class .

Proof. From the representation of , it follows that Now by (7). Thus .

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The author would like to express his gratitude to Professor Dr. V. Ravichandran and the referees for their valuable comments which have essentially improved the presentation of this paper.

1. J. Clunie and T. Sheil-Small, “Harmonic univalent functions,” Annales Academiae Scientiarum Fennicae: Series A I. Mathematica, vol. 9, pp. 3–25, 1984.
2. J. M. Jahangiri, “Harmonic functions starlike in the unit disk,” Journal of Mathematical Analysis and Applications, vol. 235, no. 2, pp. 470–477, 1999.
3. H. Silverman and E. M. Silvia, “Subclasses of harmonic univalent functions,” The New Zealand Journal of Mathematics, vol. 28, no. 2, pp. 275–284, 1999.
4. H. Silverman, “Harmonic univalent functions with negative coefficients,” Journal of Mathematical Analysis and Applications, vol. 220, no. 1, pp. 283–289, 1998.
5. O. P. Ahuja and J. M. Jahangiri, “A subclass of harmonic univalent functions,” Journal of Natural Geometry, vol. 20, no. 1-2, pp. 45–56, 2001.
6. 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.
7. R. M. El-Ashwah, M. K. Aouf, A. A. M. Hassan, and A. H. Hassan, “A unified representation of some starlike and convex harmonic functions with negative coefficients,” Opuscula Mathematica, vol. 33, no. 2, pp. 273–281, 2013.
8. B. A. Frasin, “Comprehensive family of harmonic univalent functions,” SUT Journal of Mathematics, vol. 42, no. 1, pp. 145–155, 2006.
9. J. M. Jahangiri, “Coefficient bounds and univalence criteria for harmonic functions with negative coefficients,” Annales Universitatis Mariae Curie-Skłodowska A, vol. 52, no. 2, pp. 57–66, 1998.
10. J. M. Jahangiri, G. Murugusundaramoorthy, and K. Vijaya, “Starlikeness of harmonic functions defined by Ruscheweyh derivatives,” Indian Academy of Mathematics, vol. 26, no. 1, pp. 191–200, 2004. View at: Google Scholar | MathSciNet
11. A. Janteng and S. A. Halim, “Properties of harmonic functions which are convex of order β with respect to symmetric points,” Tamkang Journal of Mathematics, vol. 40, no. 1, pp. 31–39, 2009.
12. Y. C. Kim, J. M. Jahangiri, and J. H. Choi, “Certain convex harmonic functions,” International Journal of Mathematics and Mathematical Sciences, vol. 29, no. 8, pp. 459–465, 2002.
13. N. Magesh and S. Mayilvaganan, “On a subclass of harmonic convex functions of complex order,” International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 496731, 13 pages, 2012.
14. S. Nagpal and V. Ravichandran, “A subclass of close-to-convex harmonic mappings,” Complex Variables and Elliptic Equations, vol. 59, no. 2, pp. 204–216, 2014. View at: Publisher Site | Google Scholar | MathSciNet
15. S. Nagpal and V. Ravichandran, “A comprehensive class of harmonic functions deined by convolution and its connection with integral transforms and hypergeometric functions,” Studia Universitatis Babeş-Bolyai Mathematica, vol. 59, no. 1, pp. 41–55, 2014. View at: Google Scholar
16. K. G. Subramanian, B. A. Stephen, and S. K. Lee, “Subclasses of multivalent harmonic mappings defined by convolution,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 35, no. 3, pp. 717–726, 2012.
17. M. Obradovic and S. B. Joshi, “On certain classes of strongly starlike functions,” Taiwanese Journal of Mathematics, vol. 2, no. 3, pp. 297–302, 1998.
18. K. S. Padmanabhan, “On sufficient conditions for starlikeness,” Indian Journal of Pure and Applied Mathematics, vol. 32, no. 4, pp. 543–550, 2001.
19. J.-L. Li and S. Owa, “Sufficient conditions for starlikeness,” Indian Journal of Pure and Applied Mathematics, vol. 33, no. 3, pp. 313–318, 2002.
20. N. Xu and D. Yang, “Some criteria for starlikeness and strongly starlikeness,” Bulletin of the Korean Mathematical Society, vol. 42, no. 3, pp. 579–590, 2005.
21. S. Singh and S. Gupta, “First order differential subordinations and starlikeness of analytic maps in the unit disc,” Kyungpook Mathematical Journal, vol. 45, no. 3, pp. 395–404, 2005.
22. A. Y. Lashin, “On a certain subclass of starlike functions with negative coefficients,” Journal of Inequalities in Pure and Applied Mathematics, vol. 10, no. 2, pp. 1–18, 2009.
23. H. Silverman, “Partial sums of starlike and convex functions,” Journal of Mathematical Analysis and Applications, vol. 209, no. 1, pp. 221–227, 1997.
24. A. W. Goodman, “Univalent functions and nonanalytic curves,” Proceedings of the American Mathematical Society, vol. 8, no. 3, pp. 598–601, 1957. View at: Publisher Site | Google Scholar | MathSciNet
25. S. Ruscheweyh, “Neighborhoods of univalent functions,” Proceedings of the American Mathematical Society, vol. 81, no. 4, pp. 521–527, 1981.
26. O. Altintas, Ö. Özkan, and H. M. Srivastava, “Neighborhoods of a class of analytic functions with negative coefficients,” Applied Mathematics Letters, vol. 13, no. 3, pp. 63–67, 2000.
27. S. Porwal and M. K. Aouf, “On a new subclass of harmonic univalent functions defined by fractional calculus operator,” Journal of Fractional Calculus and Applications, vol. 4, no. 10, pp. 1–12, 2013. View at: Google Scholar

#### More related articles

We are committed to sharing findings related to COVID-19 as quickly and safely as possible. Any author submitting a COVID-19 paper should notify us at help@hindawi.com to ensure their research is fast-tracked and made available on a preprint server as soon as possible. We will be providing unlimited waivers of publication charges for accepted articles related to COVID-19. Sign up here as a reviewer to help fast-track new submissions.