# Some Properties of Complex Harmonic Mapping

Authors: E. A. Eljamal; M. Darus
Journal: ISRN Applied Mathematics
 (2012)

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## 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.


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## Body

## 1. Introduction

A continuous complex-valued functionf=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
(1.1)f=h+g¯,
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 as(1.2)h(z)=z+∑n=2∞akzk,g(z)=∑n=1∞bkzk,z∈E,|b1|<1.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 forf=h+g¯ given by (1.2). For fixed positive natural m,n, and λ2≥λ1≥0,
(1.3)Dλ1,λ2m,nf(z)=Dλ1,λ2m,nh(z)+Dλ1,λ2m,ng(z),z∈E,
where
(1.4)Dλ1,λ2m,nh(z)=z+∑n=2∞(1+(λ1+λ2)(n-1)1+λ2(n-1))manzn,Dλ1,λ2m,ng(z)=∑n=1∞(1+(λ1+λ2)(n-1)1+λ2(n-1))mbnzn.We note that by specializing the parameters, especially whenλ1=λ2=0, Dλ1,λ2m,n reduces to Dm which introduced by Sălăgean in [6].LetP={(α,p)∈R2:0≤α≤1,p>0}  and Uλ1,λ2m,n(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 HSλ1,λ2m,n(α,p) the class of functions of the form (1.3) and such that
(1.5)|b1|+Uλ1,λ2m,n(α,p)(|an|+|bn|)≤1,|b1|<1.
Moreover,
(1.6)HCλ1,λ2m,n(α,p)={f∈HSλ1,λ2m,n(α,p):b1=0}.
The classes HS0,01,n(1,1)HC0,01,n(1,1), HS0,01,n(1,2)HC0,01,n(1,2) were studied in [2], and the classes HS0,01,n(1,p)HC0,01,n(1,p)(p>0) were investigated in [3]. It is known that each function of the class HC0,01,n(1,1) is starlike, and every function of the class HC0,01,n(1,2) is convex (see [2]). With respect to the following inequalities U0,01,n(1,p)=np≤Uλ1,λ2m,n(α,p)≤np+1=U0,01,n(0,p),n=2,3,…,(α,p)∈P, by condition (1.5) we have the following inclusions (1.7)HS0,01,n(0,p)⊂HSλ1,λ2m,n(α,p)⊂HS0,01,n(1,p),(α,p)∈P,HC0,01,n(0,p)⊂HCλ1,λ2m,n(α,p)⊂HC0,01,n(1,p),(α,p)∈P.


## 2. Main Result

Directly from the definition of the classHSλ1,λ2m,n(α,p)(HCλ1,λ2m,n(α,p))  we get the following.Theorem 2.1.
Let(α,p)∈P. If f∈HSλ1,λ2m,n(α,p)(HCλ1,λ2m,n(α,p)), then functions
(2.1)z⟼r-1f(rz),z⟼e-itf(eitz),z∈E,r∈(0,1),t∈R
also belong to HSλ1,λ2m,n(α,p)(HCλ1,λ2m,n(α,p)).Theorem 2.2.
If0≤α1≤α2≤1,   p>0, then
(2.2)HSλ1,λ2m,n(α1,p)⊂HSλ1,λ2m,n(α2,p),HCλ1,λ2m,n(α1,p)⊂HCλ1,λ2m,n(α2,p).
If  α∈[0,1] and 0<p1≤p2, then
(2.3)HSλ1,λ2m,n(α,p1)⊃HSλ1,λ2m,n(α,p2),HCλ1,λ2m,n(α,p1)⊃HCλ1,λ2m,n(α,p2).Theorem 2.3.
Let(α,p)∈p. If p≥1, then every function f∈HCλ1,λ2m,n(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∈HCλ1,λ2m,n(α,p) is univalent and maps the unit disk E onto a convex domain.Proof.
Ifp≥1, then Uλ1,λ2m,n(α,p)≥n for n=2,3,…,α∈[0,1], so by the condition (1.5) we obtain
(2.4)∑n=2∞n(|an|+|bn|)≤1.
Therefore (see [2]), f is univalent and starlike with respect to the origin. If p≥2, then by (1.5) we get
(2.5)∑n=2∞n2(|an|+|bn|)≤1.
Hence (see [2]), f is convex.Next, letα∈[0,1] and set p1(α)=1-log2(2-α),p2(α)=2-log2(2-α),  log21=0. We denote
(2.6)D1={(α,p)∈P:p≥p1(α)},D2={(α,p)∈P:p≥p2(α)}.
The next theorem present results concerning starlikeness and convexity of functions of the class HCλ1,λ2m,n(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 HCλ1,λ2m,n(α,p) are starlike.Proof.
We can check that the following inequality:(2.7)Uλ1,λ2m,n(α,p)≥n,(α,p)∈D1,n=2,3,…,
hold. If f∈HCλ1,λ2m,n(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∈HCλ1,λ2m,n(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∈HCλ1,λ2m,n(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 HCλ1,λ2m,n(α,p) and we have
(2.8)∑n=2∞n(|anrn-1|+|bnrn-1|)=∑n=2∞nrn-1(|an|+|bn|).
In view of properties of elementary functions, we obtain
(2.9)nrn-1≤n(r0(α,p))n-1≤Uλ1,λ2m,n(α,p),n=2,3,….
Hence, fr∈HS0,01,n(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,r0*(α,p)), where r0*(α,p)=2p-2(2-α), then each function f∈HCλ1,λ2m,n(α,p) maps the disk Er onto a convex domain.Proof.
For every(α,p)∈p∖D2 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))(2.10)n2rn-1≤Uλ1,λ2m,n(α,p),n=2,3,….
Hence fr∈HC0,01,n(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 f∈HSλ1,λ2m,n(α,p),z∈E,z≠0, then
(2.11)|f(z)|≤(1+|b1|)|z|+1-|b1|2p(2-α)|z|2,|f(z)|≥(1-|b1|)|z|-1-|b1|2p(2-α)|z|2.Proof.
Letf∈HSλ1,λ2m,n(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
(2.12)∑n=2∞(|an|+|bn|)≤1-|b1|Uλ1,λ2m,2(α,p)-∑n=3∞(Uλ1,λ2m,n(α,p)Uλ1,λ2m,2(α,p)-1)(|an|+|bn|).
Since Uλ1,λ2m,n(α,p)≥Uλ1,λ2m,2(α,p),n=3,4,…,(α,p)∈P, we have
(2.13)∑n=2∞(|an|+|bn|)≤1-|b1|Uλ1,λ2m,2(α,p).
Hence,
(2.14)|f(z)|≤∑n=2∞‍(|an|+|bn|)|z|n+(1+|b1|)|z|≤(1+|b1|)|z|+1-|b1|Uλ1,λ2m,2(α,p)|z|2,
that is, the upper estimate.
The lower estimate follows from (2.13) and the inequality:
(2.15)|f(z)|≥|z|-|b1||z|-∑n=2∞(|an|+|bn|)|z|n.Remark 2.8.
Other works related to harmonic analytic functions can be read in [7–13].



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*Source: 587689-2012-07-15.xml*