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Journal of Complex Analysis
Volume 2013 (2013), Article ID 403624, 6 pages
Subclasses of Harmonic Mappings Defined by Convolution
1Department of Mathematics, Walchand College, Sangli, Maharashtra 416415, India
2Department 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.
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.
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 .
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  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 , Silverman , and Silverman and Silvia  were amongst those who focused on the harmonic starlike functions. Later Öztürk et. al.  defined the class consisting of functions such that and are of the forms which satisfy the condition for some and for all .
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 ). (2) (see Jahangiri ). (3) (see Silverman and Silvia ). (4), with (see Avcı and Złotkiewicz  and Silverman ).(5) (see Jahangiri ). (6) (see Silverman ). (7) (see Dixit et al. ).(8) (see Ali et al. ). (9) (see Joshi et al. ).(10), where (see Jahangiri et al. ). (11), where (see Murugusundaramoorthy ). (12), where and (see Al-Shaqsi and Darus ). (13), where (see Murugusundaramoorthy et al. ).
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. , 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
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
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 .
from Theorem 5, .
Conversely, if , then and . Set , , , and .
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 .
The authors are thankful to the referees for their valuable suggestions and comments.
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