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 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=ℎ+𝑔∈𝑆𝐻, we may express the analytic functions ℎ and 𝑔 asℎ(𝑧)=𝑧+âˆžî“ğ‘›=2ğ‘Žğ‘˜ğ‘§ğ‘˜,𝑔(𝑧)=âˆžî“ğ‘›=1𝑏𝑘𝑧𝑘||𝑏,𝑧∈𝐸,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=ℎ+𝑔 given by (1.2). For fixed positive natural 𝑚,𝑛, and 𝜆2≥𝜆1≥0, 𝐷𝜆𝑚,𝑛1,𝜆2𝑓(𝑧)=𝐷𝜆𝑚,𝑛1,𝜆2ℎ(𝑧)+𝐷𝜆𝑚,𝑛1,𝜆2𝑔(𝑧),𝑧∈𝐸,(1.3) where 𝐷𝜆𝑚,𝑛1,𝜆2ℎ(𝑧)=𝑧+âˆžî“ğ‘›=2𝜆1+1+𝜆2(𝑛−1)1+𝜆2(𝑛−1)ğ‘šğ‘Žğ‘›ğ‘§ğ‘›,𝐷𝜆𝑚,𝑛1,𝜆2𝑔(𝑧)=âˆžî“ğ‘›=1𝜆1+1+𝜆2(𝑛−1)1+𝜆2(𝑛−1)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)∈𝑅2∶0≤𝛼≤1,𝑝>0}  and 𝑈𝜆𝑚,𝑛1,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 𝐻𝑆𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝) the class of functions of the form (1.3) and such that ||𝑏1||+𝑈𝜆𝑚,𝑛1,𝜆2||ğ‘Ž(𝛼,𝑝)𝑛||+||𝑏𝑛||||𝑏≤1,1||<1.(1.5) Moreover, 𝐻𝐶𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝)=𝑓∈𝐻𝑆𝜆𝑚,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)=𝑛𝑝≤𝑈𝜆𝑚,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 𝑓∈𝐻𝑆𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝)(𝐻𝐶𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝)), then functions 𝑧⟼𝑟−1𝑓(𝑟𝑧),𝑧⟼𝑒−𝑖𝑡𝑓𝑒𝑖𝑡𝑧,𝑧∈𝐸,𝑟∈(0,1),𝑡∈𝑅(2.1) also belong to 𝐻𝑆𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝)(𝐻𝐶𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝)).

Theorem 2.2. If 0≤𝛼1≤𝛼2≤1,   𝑝>0, then 𝐻𝑆𝜆𝑚,𝑛1,𝜆2𝛼1,𝑝⊂𝐻𝑆𝜆𝑚,𝑛1,𝜆2𝛼2,𝑝,𝐻𝐶𝜆𝑚,𝑛1,𝜆2𝛼1,𝑝⊂𝐻𝐶𝜆𝑚,𝑛1,𝜆2𝛼2.,𝑝(2.2) If 𝛼∈[0,1] and 0<𝑝1≤𝑝2, then 𝐻𝑆𝜆𝑚,𝑛1,𝜆2𝛼,𝑝1⊃𝐻𝑆𝜆𝑚,𝑛1,𝜆2𝛼,𝑝2,𝐻𝐶𝜆𝑚,𝑛1,𝜆2𝛼,𝑝1⊃𝐻𝐶𝜆𝑚,𝑛1,𝜆2𝛼,𝑝2.(2.3)

Theorem 2.3. Let (𝛼,𝑝)∈𝑝. If 𝑝≥1, then every function 𝑓∈𝐻𝐶𝜆𝑚,𝑛1,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∈𝐻𝐶𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝) is univalent and maps the unit disk 𝐸 onto a convex domain.

Proof. If 𝑝≥1, then 𝑈𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝)≥𝑛 for 𝑛=2,3,…,𝛼∈[0,1], so by the condition (1.5) we obtain âˆžî“ğ‘›=2𝑛||ğ‘Žğ‘›||+||𝑏𝑛||≤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 âˆžî“ğ‘›=2𝑛2||ğ‘Žğ‘›||+||𝑏𝑛||≤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 𝐷1=(𝛼,𝑝)∈𝑃∶𝑝≥𝑝1,𝐷(𝛼)2=(𝛼,𝑝)∈𝑃∶𝑝≥𝑝2(.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 𝑓∈𝐻𝐶𝜆𝑚,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)∈𝑝⧵𝐷1. If 𝑟∈(0,𝑟0(𝛼,𝑝)), where 𝑟0(𝛼,𝑝)=2𝑝−1(2−𝛼), then each function 𝑓∈𝐻𝐶𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝) maps the disk 𝐸𝑟 onto a domain starlike with respect to the origin. where 𝐸𝑟={𝑧∈𝐶∶|𝑧|<𝑟},𝑟>0,    with   𝐸1=𝐸.

Proof. For (𝛼,𝑝)∈𝑝⧵𝐷1, we have 𝑟0(𝛼,𝑝)<1, let 𝑓∈𝐻𝐶𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝), (𝛼,𝑝)∈𝑝⧵𝐷1, and let 𝑟∈(0,𝑟0(𝛼,𝑝)). By Theorem 2.1, the function 𝑓𝑟 of the form 𝑓𝑟(𝑧)=𝑟−1𝑓(𝑟𝑧) belongs to the class 𝐻𝐶𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝) and we have âˆžî“ğ‘›=2𝑛||ğ‘Žğ‘›ğ‘Ÿğ‘›âˆ’1||+||𝑏𝑛𝑟𝑛−1||=âˆžî“ğ‘›=2𝑛𝑟𝑛−1||ğ‘Žğ‘›||+||𝑏𝑛||.(2.8) In view of properties of elementary functions, we obtain 𝑛𝑟𝑛−1𝑟≤𝑛0(𝛼,𝑝)𝑛−1≤𝑈𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝),𝑛=2,3,….(2.9) Hence, 𝑓𝑟∈𝐻𝑆1,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)=2𝑝−2(2−𝛼), then each function 𝑓∈𝐻𝐶𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝) maps the disk 𝐸𝑟 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))𝑛2𝑟𝑛−1≤𝑈𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝),𝑛=2,3,….(2.10) Hence 𝑓𝑟∈𝐻𝐶1,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∈𝐻𝑆𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝),𝑧∈𝐸,𝑧≠0, then ||||≤||𝑏𝑓(𝑧)1+1||||𝑏|𝑧|+1−1||2𝑝(2−𝛼)|𝑧|2,||||≥||𝑏𝑓(𝑧)1−1||||𝑏|𝑧|−1−1||2𝑝(2−𝛼)|𝑧|2.(2.11)

Proof. Let 𝑓∈𝐻𝑆𝜆𝑚,𝑛1,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 âˆžî“ğ‘›=2||ğ‘Žğ‘›||+||𝑏𝑛||≤||𝑏1−1||𝑈𝜆𝑚,21,𝜆2(−𝛼,𝑝)âˆžî“ğ‘›=3âŽ›âŽœâŽœâŽğ‘ˆğœ†ğ‘š,𝑛1,𝜆2(𝛼,𝑝)𝑈𝜆𝑚,21,𝜆2(⎞⎟⎟⎠||ğ‘Žğ›¼,𝑝)−1𝑛||+||𝑏𝑛||.(2.12) Since 𝑈𝜆𝑚,𝑛1,𝜆2(𝛼,𝑝)≥𝑈𝜆𝑚,21,𝜆2(𝛼,𝑝),𝑛=3,4,…,(𝛼,𝑝)∈𝑃, we have âˆžî“ğ‘›=2||ğ‘Žğ‘›||+||𝑏𝑛||≤||𝑏1−1||𝑈𝜆𝑚,21,𝜆2.(𝛼,𝑝)(2.13) Hence, ||||≤𝑓(𝑧)âˆžî“ğ‘›=2||ğ‘Žğ‘›||+||𝑏𝑛|||𝑧|𝑛+||𝑏1+1|||||𝑏𝑧|≤1+1|||||𝑏𝑧|+1−1||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|âˆ’âˆžî“ğ‘›=2||ğ‘Žğ‘›||+||𝑏𝑛|||𝑧|𝑛.(2.15)

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


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