Abstract and Applied Analysis

Volume 2013 (2013), Article ID 968627, 8 pages

http://dx.doi.org/10.1155/2013/968627

## Properties of a Class of -Harmonic Functions

Department of Mathematics, Faculty of Arts and Sciences, Uludag University, 16059 Bursa, Turkey

Received 21 January 2013; Accepted 21 May 2013

Academic Editor: Youyu Wang

Copyright © 2013 Elif Yaşar and Sibel Yalçın. 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

A times continuously differentiable complex-valued function in a domain is -harmonic if satisfies the -harmonic equation , where is a positive integer. By using the generalized Salagean differential operator, we introduce a class of -harmonic functions and investigate necessary and sufficient coefficient conditions, distortion bounds, extreme points, and convex combination of the class.

#### 1. Introduction

A continuous complex-valued function in a domain is harmonic if both and are real harmonic in ; that is, and . Here represents the complex Laplacian operator

In any simply connected domain we can write , 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 in . See [1, 2].

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 functions and as

The properties of the class and its geometric subclasses have been investigated by many authors; see ([1–6]). Note that reduces to the class of normalized analytic univalent functions in if the coanalytic part of is identically zero.

A times continuously differentiable complex-valued function in a domain is -harmonic if satisfies the -harmonic equation , where is a positive integer.

A function is -harmonic in a simply connected domain if and only if has the following representation: where in for each . has the form where

Denote by the class of functions of the form (3) that are harmonic, univalent, and sense-preserving in the unit disk. Apparently, if and , is harmonic and biharmonic, respectively.

Biharmonic functions have been studied by several authors, such as, [7–9]. Also, biharmonic functions arise in many physical situations, particularly, in fluid dynamics and elasticity problems. They have many important applications in engineering, biology, and medicine, such as in [10, 11].

For a function in , differential operator was introduced by Sălăgean [12]. Al-Oboudi [13] generalized as follows: When , we get the Salagean differential operator.

For given by (2), Li and Liu [14] defined the following generalized Salagean operator in : where

For a -harmonic function given by (3), we define the following operator: If is given by (3), then from (10) we see that When , we get the generalized Salagean operator for harmonic univalent functions defined by Li and Liu [14].

Denote by the class of functions of the form (3) which satisfy the condition where is defined by (11).

We let the subclass of consist of functions of the form (3) which include , where Define .

The main object of the paper is to introduce a class of -harmonic functions by using the generalized Salagean operator which was defined by Li and Liu [14]. We investigate necessary and sufficient coefficient conditions, extreme points, distortion bounds, and convex combination of the class.

#### 2. Main Results

Theorem 1. *Let be a -harmonic function given by (3). Furthermore, let
**
where , and . Then is sense preserving, -harmonic, and univalent in and .*

*Proof. *Suppose and , so that :
which proves univalence.

In order to prove that is sense preserving, we need to show that :
for all .

Using the fact that if and only if , it suffices to show that
Substituting for in (18), we obtain
This last expression is nonnegative by (15), and so the proof is complete.

Theorem 2. *Let be given by (13) and (14). Then if and only if
**
where , , and .*

*Proof. *The “if" part follows from Theorem 1 upon noting that . For the “only if" part, we show that if the condition (20) does not hold.

Note that a necessary and sufficient condition for given by (13) and (14) to be in is that the condition (12) should be satisfied.

This is equivalent to , where
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 (20) does not hold, then the numerator in (22) is negative for is sufficiently close to . Hence there exist in for which the quotient in (22) is negative. This contradicts the required condition for and so the proof is complete.

Theorem 3. * Let be given by (13) and (14). Then if and only if
**
where
**
and , , .** In particular, the extreme points of are and , where and .*

*Proof. *For functions of the form (13) and (14) we have
Then
and so . Conversely, if , then
Set
where . Then, as required, we obtain

Theorem 4. *Let . Then for we have
*

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

The following covering result follows from the left-hand inequality in Theorem 4.

Corollary 5. * Let of the form (13) and (14) be so that . Then
*

Theorem 6. *The class is closed under convex combinations.*

*Proof. * Let for , where is given by
Then by (20),
For , , the convex combination of may be written as
Then by (34),
This is the condition required by (20) and so .

#### References

- J. Clunie and T. Sheil-Small, “Harmonic univalent functions,”
*Annales Academiae Scientiarum Fennicae. Series A I. Mathematica*, vol. 9, pp. 3–25, 1984. View at Google Scholar · View at Zentralblatt MATH · 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 - Y. Avcı and E. Złotkiewicz, “On harmonic univalent mappings,”
*Annales Universitatis Mariae Curie-Skłodowska. Sectio A*, vol. 44, pp. 1–7, 1990. View at Google Scholar · View at Zentralblatt MATH · 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 Zentralblatt MATH · View at MathSciNet · View at Scopus - 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 Zentralblatt MATH · View at MathSciNet · View at Scopus - 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 Zentralblatt MATH · View at MathSciNet - Z. Abdulhadi, Y. A. Muhanna, and S. Khuri, “On univalent solutions of the biharmonic equation,”
*Journal of Inequalities and Applications*, vol. 2005, no. 5, pp. 469–478, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Z. AbdulHadi, Y. A. Muhanna, and S. Khuri, “On some properties of solutions of the biharmonic equation,”
*Applied Mathematics and Computation*, vol. 177, no. 1, pp. 346–351, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Z. Abdulhadi and Y. Abu Muhanna, “Landau's theorem for biharmonic mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 338, no. 1, pp. 705–709, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J. Happel and H. Brenner,
*Low Reynolds Number Hydrodynamics*, Springer, New York, NY, USA, 1965. - S. A. Khuri, “Biorthogonal series solution of Stokes flow problems in sectorial regions,”
*SIAM Journal on Applied Mathematics*, vol. 56, no. 1, pp. 19–39, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - G. S. Sălăgean, “Subclasses of univalent functions,” in
*Complex Analysis—Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981)*, vol. 1013 of*Lecture Notes in Math*, pp. 362–372, Springer, Berlin, Germany, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. M. Al-Oboudi, “On univalent functions defined by a generalized Sǎlǎgean operator,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2004, no. 27, pp. 1429–1436, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. Li and P. Liu, “A new class of harmonic univalent functions by the generalized Salagean operator,”
*Wuhan University Journal of Natural Sciences*, vol. 12, no. 6, pp. 965–970, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus