Research Article | Open Access

# A New Subclass of Harmonic Univalent Functions Associated with Dziok-Srivastava Operator

**Academic Editor:**Attila Gilányi

#### Abstract

The purpose of the present paper is to study a certain subclass of harmonic univalent functions associated with Dziok-Srivastava operator. We obtain coefficient conditions, distortion bounds, and extreme points for the above class of harmonic univalent functions belonging to this class and discuss a class preserving integral operator. We also show that class studied in this paper is closed under convolution and convex combination. The results obtained for the class reduced to the corresponding results for several known classes in the literature are briefly indicated.

#### 1. Introduction

A continuous complex-valued function defined in a simply connected domain , is said to be harmonic in if both and are harmonic in . In any simply connected domain we can write , where and are analytic in . A necessary and sufficient condition for to be locally univalent and sense preserving in is that . See Clunie and Sheil-Small [1].

Denote by the class of functions that are harmonic univalent and sense preserving in the unit disk for which . Then for , we may express the analytic function and as

Note that , is reduced to the class of normalized analytic univalent functions if the coanalytic part of is identically zero.

For more basic results on harmonic univalent functions, one may refer to the following introductory text book by Duren [2] (see also [3–5]).

For and , , the generalized hypergeometric functions are defined by where is the Pochhammer symbol defined by

Corresponding to the function

The Dziok-Srivastava operator [6, 7] is defined for by

where * stands for convolution of two power series.

To make the notation simple, we write

Special cases of the Dziok-Srivastava operator includes the Hohlov operator [8], the Carlson-Shaffer operator [9], the Ruscheweyh derivative operator [10], and the Srivastava-Owa fractional derivative operators ([11–13]).

We define the Dziok-Srivastava operator of the harmonic functions given by (1.1) as

Recently, Porwal [14, Chapter 5] defined the subclass consisting of harmonic univalent functions satisfying the following condition:

He proved that if is given by (1.1) and if

For the class of is reduced to the class studied by Uralegaddi et al. [15].

Generalizing the class , we let denote the family of functions of form (1.1) which satisfy the condition where and .

Further, let be the subclass of consisting of functions of the form

In this paper, we give a sufficient condition for , given by (1.1) to be in , and it is shown that this condition is also necessary for functions in . We then obtain distortion theorem, extreme points, convolution conditions, and convex combinations and discuss a class preserving integral operator for functions in .

#### 2. Main Results

First, we give a sufficient coefficient bound for the class .

Theorem 2.1. *If is given by (1.1) and if
**
then .*

* Proof. *Let

It suffices to show that

We have
where denotes .

The last expression is bounded above by 1, if
which is equivalent to

But (2.6) is true by hypothesis.

Hence
and the theorem is proved.

In the following theorem, it is proved that the condition (2.1) is also necessary for functions that are given by (1.11).

Theorem 2.2. *A function is in , if and only if
*

*Proof. *Since , we only need to prove the “only if” part of the theorem. For this we show that if the condition (2.8) does not hold.

Note that a necessary and sufficient condition for given by (1.9) is in if
is equivalent to

The above condition must hold for all values of , . Upon choosing the values of on the positive real axis where , we must have

If the condition (2.8) does not hold then the numerator of (2.11) is negative for and sufficiently close to 1. Thus there exists a in (0,1) for which the quotient in (2.11) is negative. This contradicts the required condition for and so the proof is complete.

Next, we determine the extreme points of the closed convex hulls of , denoted by .

Theorem 2.3. *, if and only if
**
where
**, and . In particular the extreme points of are and . *

* Proof. *For functions of the form (2.12), we have

Then

and so .

Conversely, suppose that . Set , and , .

Then note that by Theorem 2.2, and . We define , and by Theorem 2.2, .

Consequently, we obtain as required.

Theorem 2.4. *If , then
*

*Proof. *We only prove the right hand inequality. The proof for left hand inequality is similar and will be omitted. Let . Taking the absolute value of , we have

For our next theorem, we need to define the convolution of two harmonic functions. For harmonic functions of the form
We define the convolution of two harmonic functions and as

Using this definition, we show that the class is closed under convolution.

Theorem 2.5. *For , let and . Then
*

*Proof. *Let be in , and be in .

Then the convolution is given by (2.19). We wish to show that the coefficients of satisfy the required condition given in Theorem 2.2. For we note that and . Now, for the convolution function , we obtain

Therefore .

Theorem 2.6. *The class is closed under convex combination. *

* Proof. *For , let , where is given by

Then by Theorem 2.2,

For , , the convex combination of may be written as

Then by (2.23),

This is the condition required by Theorem 2.2 and so .

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

Let be defined by (1.1); then defined by the relation

Theorem 3.1. *Let be given by (1.11) and , then is defined by (3.1) also belong to . *

* Proof. *Let be in , then by Theorem 2.2, we have

By definition of , we have

Now we have

Thus .

#### Acknowledgments

The first author is thankful to University Grants Commission, New Delhi (Government of India) for supporting this paper financially under research project (no. 8-2 (223)/2011 (MRP/Sc/NRCB)). The authors are also thankful to the reviewers for giving fruitful 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 | Zentralblatt MATH - P. Duren,
*Harmonic Mappings in the Plane*, vol. 156 of*Cambridge Tracts in Mathematics*, Cambridge University Press, Cambridge, UK, 2004. View at: Publisher Site - O. P. Ahuja, “Planar harmonic univalent and related mappings,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 6, no. 4, article 122, pp. 1–18, 2005. View at: Google Scholar | Zentralblatt MATH - 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 Site | Google Scholar | Zentralblatt MATH - S. Ponnusamy and A. Rasila, “Planar harmonic and quasiconformal mappings,”
*RMS Mathematics Newsletter*, vol. 17, no. 3, pp. 85–101, 2007. View at: Google Scholar - J. Dziok and H. M. Srivastava, “Certain subclasses of analytic functions associated with the generalized hypergeometric function,”
*Integral Transforms and Special Functions*, vol. 14, no. 1, pp. 7–18, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Z.-G. Wang, Y.-P. Jiang, and H. M. Srivastava, “Some subclasses of multivalent analytic functions involving the Dziok-Srivastava operator,”
*Integral Transforms and Special Functions*, vol. 19, no. 1-2, pp. 129–146, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Y. E. Hohlov, “Operators and operations on the class of univalent functions,”
*Izvestiya Vysshikh Uchebnykh Zavedeniĭ*, no. 10(197), pp. 83–89, 1978. View at: Google Scholar - B. C. Carlson and D. B. Shaffer, “Starlike and prestarlike hypergeometric functions,”
*SIAM Journal on Mathematical Analysis*, vol. 15, no. 4, pp. 737–745, 1984. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Ruscheweyh, “New criteria for univalent functions,”
*Proceedings of the American Mathematical Society*, vol. 49, pp. 109–115, 1975. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Owa, “On the distortion theorems. I,”
*Kyungpook Mathematical Journal*, vol. 18, no. 1, pp. 53–59, 1978. View at: Google Scholar | Zentralblatt MATH - S. Owa and H. M. Srivastava, “Univalent and starlike generalized hypergeometric functions,”
*Canadian Journal of Mathematics*, vol. 39, no. 5, pp. 1057–1077, 1987. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Z.-G. Wang, Y.-P. Jiang, and H. M. Srivastava, “Some subclasses of meromorphically multivalent functions associated with the generalized hypergeometric function,”
*Computers & Mathematics with Applications*, vol. 57, no. 4, pp. 571–586, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - S. Porwal,
*Study of certain classes related to analytic and harmonic univalent functions [Ph.D. thesis]*, C.S.J.M. University, Kanpur, India, 2011. - B. A. Uralegaddi, M. D. Ganigi, and S. M. Sarangi, “Univalent functions with positive coefficients,”
*Tamkang Journal of Mathematics*, vol. 25, no. 3, pp. 225–230, 1994. View at: Google Scholar | Zentralblatt MATH

#### Copyright

Copyright © 2012 A. L. Pathak et al. 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.