Research Article | Open Access
Some Properties of Complex Harmonic Mapping
We introduce new class of harmonic functions by using certain generalized differential operator of harmonic. Some results which generalize problems considered by many researchers are present. The main results are concerned with the starlikeness and convexity of certain class of harmonic functions.
A continuous complex-valued function , defined in a simply-connected complex domain , is said to be harmonic in if both and 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 ). Let be the class of functions of the form (1.1) that are harmonic univalent and sense-preserving in the unit disk for which . Then for we may express the analytic functions and as
In 1984, Clunie and Sheil-Small  investigated the class as well as its geometric subclasses and obtained some coefficient bounds. Since then, there have been several related papers on and its subclasses.
Now, we will introduce generalized derivative operator for given by (1.2). For fixed positive natural , and , where
We note that by specializing the parameters, especially when , reduces to which introduced by Sălăgean in .
Let and +, .
For a fixed pair , we denote by the class of functions of the form (1.3) and such that Moreover, The classes , were studied in , and the classes were investigated in . It is known that each function of the class is starlike, and every function of the class is convex (see ). With respect to the following inequalities , by condition (1.5) we have the following inclusions
2. Main Result
Directly from the definition of the class we get the following.
Theorem 2.1. Let . If , then functions also belong to .
Theorem 2.2. If , , then If and , then
Theorem 2.3. Let . If , then every function is univalent and maps the unit disk onto a domain starlike with respect to the origin. If , then every function is univalent and maps the unit disk onto a convex domain.
Next, let and set , . We denote The next theorem present results concerning starlikeness and convexity of functions of the class for arbitrary and , respectively.
Theorem 2.4. If , then the functions of the class are starlike.
Theorem 2.5. Let . If , where , then each function maps the disk onto a domain starlike with respect to the origin. where , with .
Proof. For , we have , let , , and let . By Theorem 2.1, the function of the form belongs to the class and we have In view of properties of elementary functions, we obtain Hence,  for any maps the onto a domain starlike with respect to the origin.
Theorem 2.6. Let . If , where , then each function maps the disk onto a convex domain.
Theorem 2.7. Let . If , then
Proof. Let , of the form (1.3) and fix . Then the condition (1.5) holds, and after simple transformations we obtain
Since , we have
that is, the upper estimate.
The lower estimate follows from (2.13) and the inequality:
The work here was supported by UKM-ST-06-FRGS0244-2010.
- T. Sheil-Small and J. Clunie, “Harmonic univalent functions,” Annales Academiae Scientiarum Fennicae A, vol. 9, pp. 3–25, 1984.
- Y. Avci and E. Zlotkiewicz, “On harmonic univalent mappings,” Annales Universitatis Mariae Curie-Skłodowska A, vol. 44, pp. 1–7, 1990.
- A. Ganczar, “On harmonic univalent mappings with small coefficients,” Demonstratio Mathematica, vol. 34, no. 3, pp. 549–558, 2001.
- A. Łazińska, “On complex mappings harmonic in the unit disc with some coefficient conditions,” Folia Scientiarum Universitatis Technicae Resoviensis, vol. 26, no. 199, pp. 107–116, 2002.
- Z. J. Jakubowski, A. Łazińska, and A. Sibelska, “On some properties of complex harmonic mappings with a two-parameter coefficient condition,” Mathematica Balkanica, vol. 18, no. 3-4, pp. 313–319, 2004.
- G. Sălăgean, “Subclasses of univalent functions,” in Complex Analysis—Fifth Romanian-Finnish Seminar, vol. 1013 of Lecture Notes in Mathematics, pp. 362–372, Springer, Berlin, Germany, 1983.
- K. Al-Shaqsi, M. Darus, and O. A. Fadipe-Joseph, “A new subclass of Salagean-type harmonic univalent functions,” Abstract and Applied Analysis, vol. 2010, Article ID 821531, 12 pages, 2010.
- M. Darus and K. Al Shaqsi, “On harmonic univalent functions defined by a generalized Ruscheweyh derivatives operator,” Lobachevskii Journal of Mathematics, vol. 22, pp. 19–26, 2006.
- K. Al-Shaqsi and M. Darus, “On generalizations of convolution for harmonic functions,” Far East Journal of Mathematical Sciences, vol. 33, no. 3, pp. 387–399, 2009.
- K. Al-Shaqsi and M. Darus, “On certain class of harmonic univalent functions,” International Journal of Contemporary Mathematical Sciences, vol. 4, no. 21–24, pp. 1193–1207, 2009.
- K. Al-Shaqsi and M. Darus, “On harmonic univalent functions with respect to k-symmetric points,” International Journal of Contemporary Mathematical Sciences, vol. 3, no. 3, pp. 111–118, 2008.
- M. Darus and K. Al-Shaqsi, “On certain subclass of harmonic univalent functions,” Journal of Analysis and Applications, vol. 6, no. 1, pp. 17–28, 2008.
- 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.
Copyright © 2012 E. A. Eljamal and M. Darus. 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.