Some Properties of Complex Harmonic Mapping

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=ℎ+𝑔,(1.1) where ℎ and 𝑔 are analytic in 𝐷. We call ℎ 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 |ℎ(𝑧)|>|𝑔(𝑧)| for all 𝑧 in 𝐷 (see [1]). Let 𝑆_𝐻 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)=𝑓_𝑧(0)−1=0. Then for 𝑓=ℎ+𝑔∈𝑆_𝐻, we may express the analytic functions ℎ and 𝑔 asℎ(𝑧)=𝑧+^∞_𝑛=2𝑎_𝑘𝑧^𝑘,𝑔(𝑧)=^∞_𝑛=1𝑏_𝑘𝑧^𝑘,𝑧∈𝐸,||𝑏₁||<1.(1.2)

In 1984, Clunie and Sheil-Small [1] investigated the class 𝑆_𝐻 as well as its geometric subclasses and obtained some coefficient bounds. Since then, there have been several related papers on 𝑆_𝐻 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=ℎ+𝑔 given by (1.2). For fixed positive natural 𝑚,𝑛, and 𝜆₂≥𝜆₁≥0, 𝐷^{𝜆𝑚,𝑛}_1,𝜆2𝑓(𝑧)=𝐷^{𝜆𝑚,𝑛}_1,𝜆2ℎ(𝑧)+𝐷^{𝜆𝑚,𝑛}_1,𝜆2𝑔(𝑧),𝑧∈𝐸,(1.3) where 𝐷^{𝜆𝑚,𝑛}_1,𝜆2ℎ(𝑧)=𝑧+^∞_𝑛=21+𝜆¹+𝜆₂(𝑛−1)1+𝜆₂(𝑛−1)^𝑚𝐷_{𝜆𝑚,𝑛1,𝜆2}𝑎_𝑛𝑧^𝑛,𝑔(𝑧)=^∞_𝑛=11+𝜆¹+𝜆₂(𝑛−1)1+𝜆₂(𝑛−1)^𝑚𝑏_𝑛𝑧^𝑛.(1.4)

We note that by specializing the parameters, especially when 𝜆₁=𝜆₂=0, 𝐷^{𝜆𝑚,𝑛}_1,𝜆2 reduces to 𝐷^𝑚 which introduced by Sălăgean in [6].

Let 𝑃={(𝛼,𝑝)∈𝑅²∶0≤𝛼≤1,𝑝>0}  and 𝑈^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝)=𝛼((1+(𝜆₁+𝜆₂)(𝑛−1))/(1+𝜆₂(𝑛−1)))^𝑚𝑝+(1−𝛼)((1+(𝜆₁+𝜆₂)(𝑛−1))/(1+𝜆₂(𝑛−1)))^𝑚(𝑝+1),  𝑛=2,3,…,(𝛼,𝑝)∈𝑃.

For a fixed pair (𝛼,𝑝)∈𝑃, we denote by 𝐻𝑆^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝) the class of functions of the form (1.3) and such that ||𝑏₁||+𝑈^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝)||𝑎_𝑛||+||𝑏_𝑛||≤1,||𝑏₁||<1.(1.5) Moreover, 𝐻𝐶^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝)=𝑓∈𝐻𝑆^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝)∶𝑏₁=0.(1.6) The classes 𝐻𝑆^1,𝑛0,0(1,1)𝐻𝐶^1,𝑛0,0(1,1), 𝐻𝑆^1,𝑛0,0(1,2)𝐻𝐶^1,𝑛0,0(1,2) were studied in [2], and the classes 𝐻𝑆^1,𝑛0,0(1,𝑝)𝐻𝐶^1,𝑛0,0(1,𝑝)(𝑝>0) were investigated in [3]. It is known that each function of the class 𝐻𝐶^1,𝑛0,0(1,1) is starlike, and every function of the class 𝐻𝐶^1,𝑛0,0(1,2) is convex (see [2]). With respect to the following inequalities 𝑈^1,𝑛0,0(1,𝑝)=𝑛^𝑝≤𝑈^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝)≤𝑛^𝑝+1=𝑈^1,𝑛0,0(0,𝑝),𝑛=2,3,…,(𝛼,𝑝)∈𝑃, by condition (1.5) we have the following inclusions 𝐻𝐻𝐶𝑆^1,𝑛0,0_1,𝑛0,0(0,𝑝)⊂𝐻𝑆^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝)⊂𝐻𝑆^1,𝑛0,0(1,𝑝),(𝛼,𝑝)∈𝑃,(0,𝑝)⊂𝐻𝐶^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝)⊂𝐻𝐶^1,𝑛0,0(1,𝑝),(𝛼,𝑝)∈𝑃.(1.7)

2. Main Result

Directly from the definition of the class 𝐻𝑆^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝)(𝐻𝐶^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝)) we get the following.

Theorem 2.1. Let (𝛼,𝑝)∈𝑃. If 𝑓∈𝐻𝑆^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝)(𝐻𝐶^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝)), then functions 𝑧⟼𝑟⁻¹𝑓(𝑟𝑧),𝑧⟼𝑒^−𝑖𝑡𝑓𝑒^𝑖𝑡𝑧,𝑧∈𝐸,𝑟∈(0,1),𝑡∈𝑅(2.1) also belong to 𝐻𝑆^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝)(𝐻𝐶^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝)).

Theorem 2.2. If 0≤𝛼₁≤𝛼₂≤1,   𝑝>0, then 𝐻𝑆^{𝜆𝑚,𝑛}_1,𝜆2𝛼₁,𝑝⊂𝐻𝑆^{𝜆𝑚,𝑛}_1,𝜆2𝛼₂,𝑝,𝐻𝐶^{𝜆𝑚,𝑛}_1,𝜆2𝛼₁,𝑝⊂𝐻𝐶^{𝜆𝑚,𝑛}_1,𝜆2𝛼₂,𝑝.(2.2) If 𝛼∈[0,1] and 0<𝑝₁≤𝑝₂, then 𝐻𝑆^{𝜆𝑚,𝑛}_1,𝜆2𝛼,𝑝₁⊃𝐻𝑆^{𝜆𝑚,𝑛}_1,𝜆2𝛼,𝑝₂,𝐻𝐶^{𝜆𝑚,𝑛}_1,𝜆2𝛼,𝑝₁⊃𝐻𝐶^{𝜆𝑚,𝑛}_1,𝜆2𝛼,𝑝₂.(2.3)

Theorem 2.3. Let (𝛼,𝑝)∈𝑝. If 𝑝≥1, then every function 𝑓∈𝐻𝐶^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝) is univalent and maps the unit disk 𝐸 onto a domain starlike with respect to the origin. If 𝑝≥2, then every function 𝑓∈𝐻𝐶^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝) is univalent and maps the unit disk 𝐸 onto a convex domain.

Proof. If 𝑝≥1, then 𝑈^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝)≥𝑛 for 𝑛=2,3,…,𝛼∈[0,1], so by the condition (1.5) we obtain ^∞_𝑛=2𝑛||𝑎_𝑛||+||𝑏_𝑛||≤1.(2.4) Therefore (see [2]), 𝑓 is univalent and starlike with respect to the origin. If 𝑝≥2, then by (1.5) we get ^∞_𝑛=2𝑛²||𝑎_𝑛||+||𝑏_𝑛||≤1.(2.5) Hence (see [2]), 𝑓 is convex.

Next, let 𝛼∈[0,1] and set 𝑝₁(𝛼)=1−log₂(2−𝛼),𝑝₂(𝛼)=2−log₂(2−𝛼),  log₂1=0. We denote 𝐷𝐷₁=(𝛼,𝑝)∈𝑃∶𝑝≥𝑝₁(𝛼)₂,=(𝛼,𝑝)∈𝑃∶𝑝≥𝑝₂(𝛼).(2.6) The next theorem present results concerning starlikeness and convexity of functions of the class 𝐻𝐶^{𝜆𝑚,𝑛}_1,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 𝐻𝐶^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝) are starlike.

Proof. We can check that the following inequality: 𝑈^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝)≥𝑛,(𝛼,𝑝)∈𝐷₁,𝑛=2,3,…,(2.7) hold. If 𝑓∈𝐻𝐶^{𝜆𝑚,𝑛}_1,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)∈𝑝⧵𝐷₁. If 𝑟∈(0,𝑟₀(𝛼,𝑝)), where 𝑟₀(𝛼,𝑝)=2^𝑝−1(2−𝛼), then each function 𝑓∈𝐻𝐶^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝) maps the disk 𝐸_𝑟 onto a domain starlike with respect to the origin. where 𝐸_𝑟={𝑧∈𝐶∶|𝑧|<𝑟},𝑟>0,    with   𝐸₁=𝐸.

Proof. For (𝛼,𝑝)∈𝑝⧵𝐷₁, we have 𝑟₀(𝛼,𝑝)<1, let 𝑓∈𝐻𝐶^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝), (𝛼,𝑝)∈𝑝⧵𝐷₁, and let 𝑟∈(0,𝑟₀(𝛼,𝑝)). By Theorem 2.1, the function 𝑓_𝑟 of the form 𝑓_𝑟(𝑧)=𝑟⁻¹𝑓(𝑟𝑧) belongs to the class 𝐻𝐶^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝) and we have ^∞_𝑛=2𝑛||𝑎_𝑛𝑟^𝑛−1||+||𝑏_𝑛𝑟^𝑛−1||=^∞_𝑛=2𝑛𝑟^𝑛−1||𝑎_𝑛||+||𝑏_𝑛||.(2.8) In view of properties of elementary functions, we obtain 𝑛𝑟^𝑛−1≤𝑛𝑟₀(𝛼,𝑝)^𝑛−1≤𝑈^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝),𝑛=2,3,….(2.9) Hence, 𝑓_𝑟∈𝐻𝑆^1,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)∈𝑝⧵𝐷₂. If 𝑟∈(0,𝑟^∗0(𝛼,𝑝)), where 𝑟^∗0(𝛼,𝑝)=2^𝑝−2(2−𝛼), then each function 𝑓∈𝐻𝐶^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝) maps the disk 𝐸_𝑟 onto a convex domain.

Proof. For every (𝛼,𝑝)∈𝑝⧵𝐷₂ we have 𝑟^∗0(𝛼,𝑝)<1. Further we proceed similarly as in the proof of Theorem 2.5, we have for any 𝑟∈(0,𝑟^∗0(𝛼,𝑝))𝑛²𝑟^𝑛−1≤𝑈^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝),𝑛=2,3,….(2.10) Hence 𝑓_𝑟∈𝐻𝐶^1,𝑛0,0(1,1) [2] for any 𝑟∈(0,𝑟^∗0(𝛼,𝑝)) maps the 𝐸 onto a convex domain.

Theorem 2.7. Let (𝛼,𝑝)∈𝑃. If 𝑓∈𝐻𝑆^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝),𝑧∈𝐸,𝑧≠0, then ||𝑓(𝑧)||≤1+||𝑏₁|||𝑧|+1−||𝑏¹||2^𝑝(2−𝛼)|𝑧|²||𝑓(𝑧)||≥1−||𝑏₁,|||𝑧|−1−||𝑏¹||2^𝑝(2−𝛼)|𝑧|².(2.11)

Proof. Let 𝑓∈𝐻𝑆^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝),(𝛼,𝑝)∈𝑃, 𝑓 of the form (1.3) and fix 𝑧∈𝐸⧵{0}. Then the condition (1.5) holds, and after simple transformations we obtain ^∞_𝑛=2||𝑎_𝑛||+||𝑏_𝑛||≤1−||𝑏¹||𝑈^𝜆𝑚,2_1,𝜆2(𝛼,𝑝)−^∞_𝑛=3⎛⎜⎜⎝𝑈^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝)𝑈^𝜆𝑚,2_1,𝜆2(𝛼,𝑝)−1⎞⎟⎟⎠||𝑎^𝑛||+||𝑏_𝑛||.(2.12) Since 𝑈^{𝜆𝑚,𝑛}_1,𝜆2(𝛼,𝑝)≥𝑈^𝜆𝑚,2_1,𝜆2(𝛼,𝑝),𝑛=3,4,…,(𝛼,𝑝)∈𝑃, we have ^∞_𝑛=2||𝑎_𝑛||+||𝑏_𝑛||≤1−||𝑏¹||𝑈^𝜆𝑚,2_1,𝜆2(𝛼,𝑝).(2.13) Hence, ||𝑓(𝑧)||≤^∞_𝑛=2||𝑎_𝑛||+||𝑏_𝑛|||𝑧|^𝑛+1+||𝑏₁|||𝑧|≤1+||𝑏₁|||𝑧|+1−||𝑏¹||𝑈^𝜆𝑚,2_1,𝜆2(𝛼,𝑝)|𝑧|²,(2.14) that is, the upper estimate.
The lower estimate follows from (2.13) and the inequality: ||𝑓(𝑧)||≥|𝑧|−||𝑏₁|||𝑧|−^∞_𝑛=2||𝑎_𝑛||+||𝑏_𝑛|||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.