About this Journal Submit a Manuscript Table of Contents
ISRN Applied Mathematics
Volume 2012 (2012), Article ID 587689, 6 pages
http://dx.doi.org/10.5402/2012/587689
Research Article

Some Properties of Complex Harmonic Mapping

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Selangor, 43600 Bangi, Malaysia

Received 10 April 2012; Accepted 11 June 2012

Academic Editors: Y. Dimakopoulos and Y.-G. Zhao

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.

Abstract

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.

1. Introduction

A continuous complex-valued function 𝑓=𝑢+𝑖𝑣, defined in a simply-connected complex domain 𝐷, is said to be harmonic in 𝐷 if both 𝑢 and 𝑣 are real harmonic in 𝐷. Such functions can be expressed as 𝑓=+𝑔,(1.1) where and 𝑔 are analytic in 𝐷. We call the analytic part and 𝑔 the coanalytic part of 𝑓. A necessary and sufficient condition for 𝑓 to be locally univalent and sense-preserving in 𝐷 is that |(𝑧)|>|𝑔(𝑧)| for all 𝑧 in 𝐷 (see [1]). Let 𝑆𝐻 be the class of functions of the form (1.1) that are harmonic univalent and sense-preserving in the unit disk 𝐸={𝑧|𝑧|<1} for which 𝑓(0)=𝑓𝑧(0)1=0. Then for 𝑓=+𝑔𝑆𝐻, we may express the analytic functions and 𝑔 as(𝑧)=𝑧+𝑛=2𝑎𝑘𝑧𝑘,𝑔(𝑧)=𝑛=1𝑏𝑘𝑧𝑘||𝑏,𝑧𝐸,1||<1.(1.2)

In 1984, Clunie and Sheil-Small [1] investigated the class 𝑆𝐻 as well as its geometric subclasses and obtained some coefficient bounds. Since then, there have been several related papers on 𝑆𝐻 and its subclasses.

In this paper, we aim at generalizing the respective results from the papers [25], that imply starlikeness and convexity of functions holomorphic in the unit disk.

Now, we will introduce generalized derivative operator for 𝑓=+𝑔 given by (1.2). For fixed positive natural 𝑚,𝑛, and 𝜆2𝜆10, 𝐷𝜆𝑚,𝑛1,𝜆2𝑓(𝑧)=𝐷𝜆𝑚,𝑛1,𝜆2(𝑧)+𝐷𝜆𝑚,𝑛1,𝜆2𝑔(𝑧),𝑧𝐸,(1.3) where 𝐷𝜆𝑚,𝑛1,𝜆2(𝑧)=𝑧+𝑛=2𝜆1+1+𝜆2(𝑛1)1+𝜆2(𝑛1)𝑚𝑎𝑛𝑧𝑛,𝐷𝜆𝑚,𝑛1,𝜆2𝑔(𝑧)=𝑛=1𝜆1+1+𝜆2(𝑛1)1+𝜆2(𝑛1)𝑚𝑏𝑛𝑧𝑛.(1.4)

We note that by specializing the parameters, especially when 𝜆1=𝜆2=0, 𝐷𝜆𝑚,𝑛1,𝜆2 reduces to 𝐷𝑚 which introduced by Sălăgean in [6].

Let 𝑃={(𝛼,𝑝)𝑅20𝛼1,𝑝>0}  and 𝑈𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝)=𝛼((1+(𝜆1+𝜆2)(𝑛1))/(1+𝜆2(𝑛1)))𝑚𝑝+(1𝛼)((1+(𝜆1+𝜆2)(𝑛1))/(1+𝜆2(𝑛1)))𝑚(𝑝+1),  𝑛=2,3,,(𝛼,𝑝)𝑃.

For a fixed pair (𝛼,𝑝)𝑃, we denote by 𝐻𝑆𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝) the class of functions of the form (1.3) and such that ||𝑏1||+𝑈𝜆𝑚,𝑛1,𝜆2||𝑎(𝛼,𝑝)𝑛||+||𝑏𝑛||||𝑏1,1||<1.(1.5) Moreover, 𝐻𝐶𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝)=𝑓𝐻𝑆𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝)𝑏1=0.(1.6) The classes 𝐻𝑆1,𝑛0,0(1,1)𝐻𝐶1,𝑛0,0(1,1), 𝐻𝑆1,𝑛0,0(1,2)𝐻𝐶1,𝑛0,0(1,2) were studied in [2], and the classes 𝐻𝑆1,𝑛0,0(1,𝑝)𝐻𝐶1,𝑛0,0(1,𝑝)(𝑝>0) were investigated in [3]. It is known that each function of the class 𝐻𝐶1,𝑛0,0(1,1) is starlike, and every function of the class 𝐻𝐶1,𝑛0,0(1,2) is convex (see [2]). With respect to the following inequalities 𝑈1,𝑛0,0(1,𝑝)=𝑛𝑝𝑈𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝)𝑛𝑝+1=𝑈1,𝑛0,0(0,𝑝),𝑛=2,3,,(𝛼,𝑝)𝑃, by condition (1.5) we have the following inclusions 𝐻𝑆1,𝑛0,0(0,𝑝)𝐻𝑆𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝)𝐻𝑆1,𝑛0,0(1,𝑝),(𝛼,𝑝)𝑃,𝐻𝐶1,𝑛0,0(0,𝑝)𝐻𝐶𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝)𝐻𝐶1,𝑛0,0(1,𝑝),(𝛼,𝑝)𝑃.(1.7)

2. Main Result

Directly from the definition of the class 𝐻𝑆𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝)(𝐻𝐶𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝)) we get the following.

Theorem 2.1. Let (𝛼,𝑝)𝑃. If 𝑓𝐻𝑆𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝)(𝐻𝐶𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝)), then functions 𝑧𝑟1𝑓(𝑟𝑧),𝑧𝑒𝑖𝑡𝑓𝑒𝑖𝑡𝑧,𝑧𝐸,𝑟(0,1),𝑡𝑅(2.1) also belong to 𝐻𝑆𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝)(𝐻𝐶𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝)).

Theorem 2.2. If 0𝛼1𝛼21,   𝑝>0, then 𝐻𝑆𝜆𝑚,𝑛1,𝜆2𝛼1,𝑝𝐻𝑆𝜆𝑚,𝑛1,𝜆2𝛼2,𝑝,𝐻𝐶𝜆𝑚,𝑛1,𝜆2𝛼1,𝑝𝐻𝐶𝜆𝑚,𝑛1,𝜆2𝛼2.,𝑝(2.2) If 𝛼[0,1] and 0<𝑝1𝑝2, then 𝐻𝑆𝜆𝑚,𝑛1,𝜆2𝛼,𝑝1𝐻𝑆𝜆𝑚,𝑛1,𝜆2𝛼,𝑝2,𝐻𝐶𝜆𝑚,𝑛1,𝜆2𝛼,𝑝1𝐻𝐶𝜆𝑚,𝑛1,𝜆2𝛼,𝑝2.(2.3)

Theorem 2.3. Let (𝛼,𝑝)𝑝. If 𝑝1, then every function 𝑓𝐻𝐶𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝) is univalent and maps the unit disk 𝐸 onto a domain starlike with respect to the origin. If 𝑝2, then every function 𝑓𝐻𝐶𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝) is univalent and maps the unit disk 𝐸 onto a convex domain.

Proof. If 𝑝1, then 𝑈𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝)𝑛 for 𝑛=2,3,,𝛼[0,1], so by the condition (1.5) we obtain 𝑛=2𝑛||𝑎𝑛||+||𝑏𝑛||1.(2.4) Therefore (see [2]), 𝑓 is univalent and starlike with respect to the origin. If 𝑝2, then by (1.5) we get 𝑛=2𝑛2||𝑎𝑛||+||𝑏𝑛||1.(2.5) Hence (see [2]), 𝑓 is convex.

Next, let 𝛼[0,1] and set 𝑝1(𝛼)=1log2(2𝛼),𝑝2(𝛼)=2log2(2𝛼),  log21=0. We denote 𝐷1=(𝛼,𝑝)𝑃𝑝𝑝1,𝐷(𝛼)2=(𝛼,𝑝)𝑃𝑝𝑝2(.𝛼)(2.6) The next theorem present results concerning starlikeness and convexity of functions of the class 𝐻𝐶𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝) for arbitrary (𝛼,𝑝)𝐷1 and (𝛼,𝑝)𝐷2, respectively.

Theorem 2.4. If (𝛼,𝑝)𝐷1, then the functions of the class 𝐻𝐶𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝) are starlike.

Proof. We can check that the following inequality: 𝑈𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝)𝑛,(𝛼,𝑝)𝐷1,𝑛=2,3,,(2.7) hold. If 𝑓𝐻𝐶𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝) for (𝛼,𝑝)𝐷1, then in view of the inequality, the condition (1.5) and of the mentioned result from [2] it follows that 𝑓 is a starlike function.

Theorem 2.5. Let (𝛼,𝑝)𝑝𝐷1. If 𝑟(0,𝑟0(𝛼,𝑝)), where 𝑟0(𝛼,𝑝)=2𝑝1(2𝛼), then each function 𝑓𝐻𝐶𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝) maps the disk 𝐸𝑟 onto a domain starlike with respect to the origin. where 𝐸𝑟={𝑧𝐶|𝑧|<𝑟},𝑟>0,    with   𝐸1=𝐸.

Proof. For (𝛼,𝑝)𝑝𝐷1, we have 𝑟0(𝛼,𝑝)<1, let 𝑓𝐻𝐶𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝), (𝛼,𝑝)𝑝𝐷1, and let 𝑟(0,𝑟0(𝛼,𝑝)). By Theorem 2.1, the function 𝑓𝑟 of the form 𝑓𝑟(𝑧)=𝑟1𝑓(𝑟𝑧) belongs to the class 𝐻𝐶𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝) and we have 𝑛=2𝑛||𝑎𝑛𝑟𝑛1||+||𝑏𝑛𝑟𝑛1||=𝑛=2𝑛𝑟𝑛1||𝑎𝑛||+||𝑏𝑛||.(2.8) In view of properties of elementary functions, we obtain 𝑛𝑟𝑛1𝑟𝑛0(𝛼,𝑝)𝑛1𝑈𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝),𝑛=2,3,.(2.9) Hence, 𝑓𝑟𝐻𝑆1,𝑛0,0(1,1) [2] for any 𝑟(0,𝑟0(𝛼,𝑝)) maps the 𝐸 onto a domain starlike with respect to the origin.

Theorem 2.6. Let (𝛼,𝑝)𝑝𝐷2. If 𝑟(0,𝑟0(𝛼,𝑝)), where 𝑟0(𝛼,𝑝)=2𝑝2(2𝛼), then each function 𝑓𝐻𝐶𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝) maps the disk 𝐸𝑟 onto a convex domain.

Proof. For every (𝛼,𝑝)𝑝𝐷2 we have 𝑟0(𝛼,𝑝)<1. Further we proceed similarly as in the proof of Theorem 2.5, we have for any 𝑟(0,𝑟0(𝛼,𝑝))𝑛2𝑟𝑛1𝑈𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝),𝑛=2,3,.(2.10) Hence 𝑓𝑟𝐻𝐶1,𝑛0,0(1,1) [2] for any 𝑟(0,𝑟0(𝛼,𝑝)) maps the 𝐸 onto a convex domain.

Theorem 2.7. Let (𝛼,𝑝)𝑃. If 𝑓𝐻𝑆𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝),𝑧𝐸,𝑧0, then ||||||𝑏𝑓(𝑧)1+1||||𝑏|𝑧|+11||2𝑝(2𝛼)|𝑧|2,||||||𝑏𝑓(𝑧)11||||𝑏|𝑧|11||2𝑝(2𝛼)|𝑧|2.(2.11)

Proof. Let 𝑓𝐻𝑆𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝),(𝛼,𝑝)𝑃, 𝑓 of the form (1.3) and fix 𝑧𝐸{0}. Then the condition (1.5) holds, and after simple transformations we obtain 𝑛=2||𝑎𝑛||+||𝑏𝑛||||𝑏11||𝑈𝜆𝑚,21,𝜆2(𝛼,𝑝)𝑛=3𝑈𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝)𝑈𝜆𝑚,21,𝜆2(||𝑎𝛼,𝑝)1𝑛||+||𝑏𝑛||.(2.12) Since 𝑈𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝)𝑈𝜆𝑚,21,𝜆2(𝛼,𝑝),𝑛=3,4,,(𝛼,𝑝)𝑃, we have 𝑛=2||𝑎𝑛||+||𝑏𝑛||||𝑏11||𝑈𝜆𝑚,21,𝜆2.(𝛼,𝑝)(2.13) Hence, ||||𝑓(𝑧)𝑛=2||𝑎𝑛||+||𝑏𝑛|||𝑧|𝑛+||𝑏1+1|||||𝑏𝑧|1+1|||||𝑏𝑧|+11||𝑈𝜆𝑚,21,𝜆2|(𝛼,𝑝)𝑧|2,(2.14) that is, the upper estimate.
The lower estimate follows from (2.13) and the inequality: ||||||𝑏𝑓(𝑧)|𝑧|1|||𝑧|𝑛=2||𝑎𝑛||+||𝑏𝑛|||𝑧|𝑛.(2.15)

Remark 2.8. Other works related to harmonic analytic functions can be read in [713].

Acknowledgment

The work here was supported by UKM-ST-06-FRGS0244-2010.

References

  1. T. Sheil-Small and J. Clunie, “Harmonic univalent functions,” Annales Academiae Scientiarum Fennicae A, vol. 9, pp. 3–25, 1984.
  2. Y. Avci and E. Zlotkiewicz, “On harmonic univalent mappings,” Annales Universitatis Mariae Curie-Skłodowska A, vol. 44, pp. 1–7, 1990.
  3. A. Ganczar, “On harmonic univalent mappings with small coefficients,” Demonstratio Mathematica, vol. 34, no. 3, pp. 549–558, 2001.
  4. 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.
  5. 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.
  6. 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. View at Publisher · View at Google Scholar
  7. 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. View at Publisher · View at Google Scholar
  8. 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.
  9. 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.
  10. 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.
  11. 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.
  12. 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.
  13. 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. View at Publisher · View at Google Scholar