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International Scholarly Research Notices / 2012 / Article

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Volume 2012 |Article ID 587689 | https://doi.org/10.5402/2012/587689

E. A. Eljamal, M. Darus, "Some Properties of Complex Harmonic Mapping", International Scholarly Research Notices, vol. 2012, Article ID 587689, 6 pages, 2012. https://doi.org/10.5402/2012/587689

Some Properties of Complex Harmonic Mapping

E. A. Eljamal1 and M. Darus1

1School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Selangor, 43600 Bangi, Malaysia

Academic Editor: Y. Dimakopoulos
Received10 Apr 2012
Accepted11 Jun 2012
Published15 Jul 2012

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=𝑒+𝑖𝑣, defined in a simply-connected complex domain 𝐷, is said to be harmonic in 𝐷 if both 𝑒 and 𝑣 are real harmonic in 𝐷. Such functions can be expressed as 𝑓=β„Ž+𝑔,(1.1) where β„Ž and 𝑔 are analytic in 𝐷. We call β„Ž 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 |β„Ž(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={π‘§βˆΆ|𝑧|<1} for which 𝑓(0)=𝑓𝑧(0)βˆ’1=0. Then for 𝑓=β„Ž+π‘”βˆˆπ‘†π», we may express the analytic functions β„Ž and 𝑔 asβ„Ž(𝑧)=𝑧+βˆžξ“π‘›=2π‘Žπ‘˜π‘§π‘˜,𝑔(𝑧)=βˆžξ“π‘›=1π‘π‘˜π‘§π‘˜||𝑏,π‘§βˆˆπΈ,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 [2–5], that imply starlikeness and convexity of functions holomorphic in the unit disk.

Now, we will introduce generalized derivative operator for 𝑓=β„Ž+𝑔 given by (1.2). For fixed positive natural π‘š,𝑛, and πœ†2β‰₯πœ†1β‰₯0, π·πœ†π‘š,𝑛1,πœ†2𝑓(𝑧)=π·πœ†π‘š,𝑛1,πœ†2β„Ž(𝑧)+π·πœ†π‘š,𝑛1,πœ†2𝑔(𝑧),π‘§βˆˆπΈ,(1.3) where π·πœ†π‘š,𝑛1,πœ†2β„Ž(𝑧)=𝑧+βˆžξ“π‘›=2ξƒ©ξ€·πœ†1+1+πœ†2ξ€Έ(π‘›βˆ’1)1+πœ†2ξƒͺ(π‘›βˆ’1)π‘šπ‘Žπ‘›π‘§π‘›,π·πœ†π‘š,𝑛1,πœ†2𝑔(𝑧)=βˆžξ“π‘›=1ξƒ©ξ€·πœ†1+1+πœ†2ξ€Έ(π‘›βˆ’1)1+πœ†2ξƒͺ(π‘›βˆ’1)π‘šπ‘π‘›π‘§π‘›.(1.4)

We note that by specializing the parameters, especially when πœ†1=πœ†2=0, π·πœ†π‘š,𝑛1,πœ†2 reduces to π·π‘š which introduced by SΔƒlΔƒgean in [6].

Let 𝑃={(𝛼,𝑝)βˆˆπ‘…2∢0≀𝛼≀1,𝑝>0}  and π‘ˆπœ†π‘š,𝑛1,πœ†2(𝛼,𝑝)=𝛼((1+(πœ†1+πœ†2)(π‘›βˆ’1))/(1+πœ†2(π‘›βˆ’1)))π‘šπ‘+(1βˆ’π›Ό)((1+(πœ†1+πœ†2)(π‘›βˆ’1))/(1+πœ†2(π‘›βˆ’1)))π‘š(𝑝+1),  𝑛=2,3,…,(𝛼,𝑝)βˆˆπ‘ƒ.

For a fixed pair (𝛼,𝑝)βˆˆπ‘ƒ, we denote by π»π‘†πœ†π‘š,𝑛1,πœ†2(𝛼,𝑝) the class of functions of the form (1.3) and such that ||𝑏1||+π‘ˆπœ†π‘š,𝑛1,πœ†2ξ€·||π‘Ž(𝛼,𝑝)𝑛||+||𝑏𝑛||ξ€Έ||𝑏≀1,1||<1.(1.5) Moreover, π»πΆπœ†π‘š,𝑛1,πœ†2(𝛼,𝑝)=π‘“βˆˆπ»π‘†πœ†π‘š,𝑛1,πœ†2(𝛼,𝑝)βˆΆπ‘1=0.(1.6) The classes 𝐻𝑆1,𝑛0,0(1,1)𝐻𝐢1,𝑛0,0(1,1), 𝐻𝑆1,𝑛0,0(1,2)𝐻𝐢1,𝑛0,0(1,2) were studied in [2], and the classes 𝐻𝑆1,𝑛0,0(1,𝑝)𝐻𝐢1,𝑛0,0(1,𝑝)(𝑝>0) were investigated in [3]. It is known that each function of the class 𝐻𝐢1,𝑛0,0(1,1) is starlike, and every function of the class 𝐻𝐢1,𝑛0,0(1,2) is convex (see [2]). With respect to the following inequalities π‘ˆ1,𝑛0,0(1,𝑝)=π‘›π‘β‰€π‘ˆπœ†π‘š,𝑛1,πœ†2(𝛼,𝑝)≀𝑛𝑝+1=π‘ˆ1,𝑛0,0(0,𝑝),𝑛=2,3,…,(𝛼,𝑝)βˆˆπ‘ƒ, by condition (1.5) we have the following inclusions 𝐻𝑆1,𝑛0,0(0,𝑝)βŠ‚π»π‘†πœ†π‘š,𝑛1,πœ†2(𝛼,𝑝)βŠ‚π»π‘†1,𝑛0,0(1,𝑝),(𝛼,𝑝)βˆˆπ‘ƒ,𝐻𝐢1,𝑛0,0(0,𝑝)βŠ‚π»πΆπœ†π‘š,𝑛1,πœ†2(𝛼,𝑝)βŠ‚π»πΆ1,𝑛0,0(1,𝑝),(𝛼,𝑝)βˆˆπ‘ƒ.(1.7)

2. Main Result

Directly from the definition of the class π»π‘†πœ†π‘š,𝑛1,πœ†2(𝛼,𝑝)(π»πΆπœ†π‘š,𝑛1,πœ†2(𝛼,𝑝)) we get the following.

Theorem 2.1. Let (𝛼,𝑝)βˆˆπ‘ƒ. If π‘“βˆˆπ»π‘†πœ†π‘š,𝑛1,πœ†2(𝛼,𝑝)(π»πΆπœ†π‘š,𝑛1,πœ†2(𝛼,𝑝)), then functions π‘§βŸΌπ‘Ÿβˆ’1𝑓(π‘Ÿπ‘§),π‘§βŸΌπ‘’βˆ’π‘–π‘‘π‘“ξ€·π‘’π‘–π‘‘π‘§ξ€Έ,π‘§βˆˆπΈ,π‘Ÿβˆˆ(0,1),π‘‘βˆˆπ‘…(2.1) also belong to π»π‘†πœ†π‘š,𝑛1,πœ†2(𝛼,𝑝)(π»πΆπœ†π‘š,𝑛1,πœ†2(𝛼,𝑝)).

Theorem 2.2. If 0≀𝛼1≀𝛼2≀1,   𝑝>0, then π»π‘†πœ†π‘š,𝑛1,πœ†2𝛼1ξ€Έ,π‘βŠ‚π»π‘†πœ†π‘š,𝑛1,πœ†2𝛼2ξ€Έ,𝑝,π»πΆπœ†π‘š,𝑛1,πœ†2𝛼1ξ€Έ,π‘βŠ‚π»πΆπœ†π‘š,𝑛1,πœ†2𝛼2ξ€Έ.,𝑝(2.2) If π›Όβˆˆ[0,1] and 0<𝑝1≀𝑝2, then π»π‘†πœ†π‘š,𝑛1,πœ†2𝛼,𝑝1ξ€ΈβŠƒπ»π‘†πœ†π‘š,𝑛1,πœ†2𝛼,𝑝2ξ€Έ,π»πΆπœ†π‘š,𝑛1,πœ†2𝛼,𝑝1ξ€ΈβŠƒπ»πΆπœ†π‘š,𝑛1,πœ†2𝛼,𝑝2ξ€Έ.(2.3)

Theorem 2.3. Let (𝛼,𝑝)βˆˆπ‘. If 𝑝β‰₯1, then every function π‘“βˆˆπ»πΆπœ†π‘š,𝑛1,πœ†2(𝛼,𝑝) is univalent and maps the unit disk 𝐸 onto a domain starlike with respect to the origin. If 𝑝β‰₯2, then every function π‘“βˆˆπ»πΆπœ†π‘š,𝑛1,πœ†2(𝛼,𝑝) is univalent and maps the unit disk 𝐸 onto a convex domain.

Proof. If 𝑝β‰₯1, then π‘ˆπœ†π‘š,𝑛1,πœ†2(𝛼,𝑝)β‰₯𝑛 for 𝑛=2,3,…,π›Όβˆˆ[0,1], so by the condition (1.5) we obtain βˆžξ“π‘›=2𝑛||π‘Žπ‘›||+||𝑏𝑛||≀1.(2.4) Therefore (see [2]), 𝑓 is univalent and starlike with respect to the origin. If 𝑝β‰₯2, then by (1.5) we get βˆžξ“π‘›=2𝑛2ξ€·||π‘Žπ‘›||+||𝑏𝑛||≀1.(2.5) Hence (see [2]), 𝑓 is convex.

Next, let π›Όβˆˆ[0,1] and set 𝑝1(𝛼)=1βˆ’log2(2βˆ’π›Ό),𝑝2(𝛼)=2βˆ’log2(2βˆ’π›Ό),  log21=0. We denote 𝐷1=ξ€½(𝛼,𝑝)βˆˆπ‘ƒβˆΆπ‘β‰₯𝑝1ξ€Ύ,𝐷(𝛼)2=ξ€½(𝛼,𝑝)βˆˆπ‘ƒβˆΆπ‘β‰₯𝑝2(ξ€Ύ.𝛼)(2.6) The next theorem present results concerning starlikeness and convexity of functions of the class π»πΆπœ†π‘š,𝑛1,πœ†2(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 π»πΆπœ†π‘š,𝑛1,πœ†2(𝛼,𝑝) are starlike.

Proof. We can check that the following inequality: π‘ˆπœ†π‘š,𝑛1,πœ†2(𝛼,𝑝)β‰₯𝑛,(𝛼,𝑝)∈𝐷1,𝑛=2,3,…,(2.7) hold. If π‘“βˆˆπ»πΆπœ†π‘š,𝑛1,πœ†2(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)βˆˆπ‘β§΅π·1. If π‘Ÿβˆˆ(0,π‘Ÿ0(𝛼,𝑝)), where π‘Ÿ0(𝛼,𝑝)=2π‘βˆ’1(2βˆ’π›Ό), then each function π‘“βˆˆπ»πΆπœ†π‘š,𝑛1,πœ†2(𝛼,𝑝) maps the disk πΈπ‘Ÿ onto a domain starlike with respect to the origin. where πΈπ‘Ÿ={π‘§βˆˆπΆβˆΆ|𝑧|<π‘Ÿ},π‘Ÿ>0,    with   𝐸1=𝐸.

Proof. For (𝛼,𝑝)βˆˆπ‘β§΅π·1, we have π‘Ÿ0(𝛼,𝑝)<1, let π‘“βˆˆπ»πΆπœ†π‘š,𝑛1,πœ†2(𝛼,𝑝), (𝛼,𝑝)βˆˆπ‘β§΅π·1, and let π‘Ÿβˆˆ(0,π‘Ÿ0(𝛼,𝑝)). By Theorem 2.1, the function π‘“π‘Ÿ of the form π‘“π‘Ÿ(𝑧)=π‘Ÿβˆ’1𝑓(π‘Ÿπ‘§) belongs to the class π»πΆπœ†π‘š,𝑛1,πœ†2(𝛼,𝑝) and we have βˆžξ“π‘›=2𝑛||π‘Žπ‘›π‘Ÿπ‘›βˆ’1||+||π‘π‘›π‘Ÿπ‘›βˆ’1||ξ€Έ=βˆžξ“π‘›=2π‘›π‘Ÿπ‘›βˆ’1ξ€·||π‘Žπ‘›||+||𝑏𝑛||ξ€Έ.(2.8) In view of properties of elementary functions, we obtain π‘›π‘Ÿπ‘›βˆ’1ξ€·π‘Ÿβ‰€π‘›0ξ€Έ(𝛼,𝑝)π‘›βˆ’1β‰€π‘ˆπœ†π‘š,𝑛1,πœ†2(𝛼,𝑝),𝑛=2,3,….(2.9) Hence, π‘“π‘Ÿβˆˆπ»π‘†1,𝑛0,0(1,1) [2] for any π‘Ÿβˆˆ(0,π‘Ÿ0(𝛼,𝑝)) maps the 𝐸 onto a domain starlike with respect to the origin.

Theorem 2.6. Let (𝛼,𝑝)βˆˆπ‘β§΅π·2. If π‘Ÿβˆˆ(0,π‘Ÿβˆ—0(𝛼,𝑝)), where π‘Ÿβˆ—0(𝛼,𝑝)=2π‘βˆ’2(2βˆ’π›Ό), then each function π‘“βˆˆπ»πΆπœ†π‘š,𝑛1,πœ†2(𝛼,𝑝) maps the disk πΈπ‘Ÿ onto a convex domain.

Proof. For every (𝛼,𝑝)βˆˆπ‘β§΅π·2 we have π‘Ÿβˆ—0(𝛼,𝑝)<1. Further we proceed similarly as in the proof of Theorem 2.5, we have for any π‘Ÿβˆˆ(0,π‘Ÿβˆ—0(𝛼,𝑝))𝑛2π‘Ÿπ‘›βˆ’1β‰€π‘ˆπœ†π‘š,𝑛1,πœ†2(𝛼,𝑝),𝑛=2,3,….(2.10) Hence π‘“π‘Ÿβˆˆπ»πΆ1,𝑛0,0(1,1) [2] for any π‘Ÿβˆˆ(0,π‘Ÿβˆ—0(𝛼,𝑝)) maps the 𝐸 onto a convex domain.

Theorem 2.7. Let (𝛼,𝑝)βˆˆπ‘ƒ. If π‘“βˆˆπ»π‘†πœ†π‘š,𝑛1,πœ†2(𝛼,𝑝),π‘§βˆˆπΈ,𝑧≠0, then ||||≀||𝑏𝑓(𝑧)1+1||ξ€Έ||𝑏|𝑧|+1βˆ’1||2𝑝(2βˆ’π›Ό)|𝑧|2,||||β‰₯ξ€·||𝑏𝑓(𝑧)1βˆ’1||ξ€Έ||𝑏|𝑧|βˆ’1βˆ’1||2𝑝(2βˆ’π›Ό)|𝑧|2.(2.11)

Proof. Let π‘“βˆˆπ»π‘†πœ†π‘š,𝑛1,πœ†2(𝛼,𝑝),(𝛼,𝑝)βˆˆπ‘ƒ, 𝑓 of the form (1.3) and fix π‘§βˆˆπΈβ§΅{0}. Then the condition (1.5) holds, and after simple transformations we obtain βˆžξ“π‘›=2ξ€·||π‘Žπ‘›||+||𝑏𝑛||≀||𝑏1βˆ’1||π‘ˆπœ†π‘š,21,πœ†2(βˆ’π›Ό,𝑝)βˆžξ“π‘›=3βŽ›βŽœβŽœβŽπ‘ˆπœ†π‘š,𝑛1,πœ†2(𝛼,𝑝)π‘ˆπœ†π‘š,21,πœ†2(βŽžβŽŸβŽŸβŽ ξ€·||π‘Žπ›Ό,𝑝)βˆ’1𝑛||+||𝑏𝑛||ξ€Έ.(2.12) Since π‘ˆπœ†π‘š,𝑛1,πœ†2(𝛼,𝑝)β‰₯π‘ˆπœ†π‘š,21,πœ†2(𝛼,𝑝),𝑛=3,4,…,(𝛼,𝑝)βˆˆπ‘ƒ, we have βˆžξ“π‘›=2ξ€·||π‘Žπ‘›||+||𝑏𝑛||≀||𝑏1βˆ’1||π‘ˆπœ†π‘š,21,πœ†2.(𝛼,𝑝)(2.13) Hence, ||||≀𝑓(𝑧)βˆžξ“π‘›=2ξ€·||π‘Žπ‘›||+||𝑏𝑛||ξ€Έ|𝑧|𝑛+ξ€·||𝑏1+1||ξ€Έ|ξ€·||𝑏𝑧|≀1+1||ξ€Έ|||𝑏𝑧|+1βˆ’1||π‘ˆπœ†π‘š,21,πœ†2|(𝛼,𝑝)𝑧|2,(2.14) that is, the upper estimate.
The lower estimate follows from (2.13) and the inequality: ||||||𝑏𝑓(𝑧)β‰₯|𝑧|βˆ’1|||𝑧|βˆ’βˆžξ“π‘›=2ξ€·||π‘Žπ‘›||+||𝑏𝑛||ξ€Έ|𝑧|𝑛.(2.15)

Remark 2.8. Other works related to harmonic analytic functions can be read in [7–13].

Acknowledgment

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

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