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ISRN Applied Mathematics
Volume 2012 (2012), Article ID 587689, 6 pages
doi:10.5402/2012/587689
Research Article

Some Properties of Complex Harmonic Mapping

E. A. Eljamal and M. Darus

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 + 𝑖 𝑣 , 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 as ℎ ( 𝑧 ) = 𝑧 + ∞ 𝑛 = 2 𝑎 𝑘 𝑧 𝑘 , 𝑔 ( 𝑧 ) = ∞ 𝑛 = 1 𝑏 𝑘 𝑧 𝑘 | | 𝑏 , 𝑧 ∈ 𝐸 , 1 | | < 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 𝜆 2 ≥ 𝜆 1 ≥ 0 , 𝐷 𝜆 𝑚 , 𝑛 1 , 𝜆 2 𝑓 ( 𝑧 ) = 𝐷 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ℎ ( 𝑧 ) + 𝐷 𝜆 𝑚 , 𝑛 1 , 𝜆 2 𝑔 ( 𝑧 ) , 𝑧 ∈ 𝐸 , ( 1 . 3 ) where 𝐷 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ℎ ( 𝑧 ) = 𝑧 + ∞ 𝑛 = 2 𝜆 1 + 1 + 𝜆 2 ( 𝑛 − 1 ) 1 + 𝜆 2 ( 𝑛 − 1 ) 𝑚 𝑎 𝑛 𝑧 𝑛 , 𝐷 𝜆 𝑚 , 𝑛 1 , 𝜆 2 𝑔 ( 𝑧 ) = ∞ 𝑛 = 1 𝜆 1 + 1 + 𝜆 2 ( 𝑛 − 1 ) 1 + 𝜆 2 ( 𝑛 − 1 ) 𝑚 𝑏 𝑛 𝑧 𝑛 . ( 1 . 4 )

We note that by specializing the parameters, especially when 𝜆 1 = 𝜆 2 = 0 , 𝐷 𝜆 𝑚 , 𝑛 1 , 𝜆 2 reduces to 𝐷 𝑚 which introduced by Sălăgean in [6].

Let 𝑃 = { ( 𝛼 , 𝑝 ) ∈ 𝑅 2 ∶ 0 ≤ 𝛼 ≤ 1 , 𝑝 > 0 }   and 𝑈 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) = 𝛼 ( ( 1 + ( 𝜆 1 + 𝜆 2 ) ( 𝑛 − 1 ) ) / ( 1 + 𝜆 2 ( 𝑛 − 1 ) ) ) 𝑚 𝑝 + ( 1 − 𝛼 ) ( ( 1 + ( 𝜆 1 + 𝜆 2 ) ( 𝑛 − 1 ) ) / ( 1 + 𝜆 2 ( 𝑛 − 1 ) ) ) 𝑚 ( 𝑝 + 1 ) ,   𝑛 = 2 , 3 , … , ( 𝛼 , 𝑝 ) ∈ 𝑃 .

For a fixed pair ( 𝛼 , 𝑝 ) ∈ 𝑃 , we denote by 𝐻 𝑆 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) the class of functions of the form (1.3) and such that | | 𝑏 1 | | + 𝑈 𝜆 𝑚 , 𝑛 1 , 𝜆 2 | | 𝑎 ( 𝛼 , 𝑝 ) 𝑛 | | + | | 𝑏 𝑛 | | | | 𝑏 ≤ 1 , 1 | | < 1 . ( 1 . 5 ) Moreover, 𝐻 𝐶 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) = 𝑓 ∈ 𝐻 𝑆 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) ∶ 𝑏 1 = 0 . ( 1 . 6 ) The classes 𝐻 𝑆 1 , 𝑛 0 , 0 ( 1 , 1 ) 𝐻 𝐶 1 , 𝑛 0 , 0 ( 1 , 1 ) , 𝐻 𝑆 1 , 𝑛 0 , 0 ( 1 , 2 ) 𝐻 𝐶 1 , 𝑛 0 , 0 ( 1 , 2 ) were studied in [2], and the classes 𝐻 𝑆 1 , 𝑛 0 , 0 ( 1 , 𝑝 ) 𝐻 𝐶 1 , 𝑛 0 , 0 ( 1 , 𝑝 ) ( 𝑝 > 0 ) were investigated in [3]. It is known that each function of the class 𝐻 𝐶 1 , 𝑛 0 , 0 ( 1 , 1 ) is starlike, and every function of the class 𝐻 𝐶 1 , 𝑛 0 , 0 ( 1 , 2 ) is convex (see [2]). With respect to the following inequalities 𝑈 1 , 𝑛 0 , 0 ( 1 , 𝑝 ) = 𝑛 𝑝 ≤ 𝑈 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) ≤ 𝑛 𝑝 + 1 = 𝑈 1 , 𝑛 0 , 0 ( 0 , 𝑝 ) , 𝑛 = 2 , 3 , … , ( 𝛼 , 𝑝 ) ∈ 𝑃 , by condition (1.5) we have the following inclusions 𝐻 𝑆 1 , 𝑛 0 , 0 ( 0 , 𝑝 ) ⊂ 𝐻 𝑆 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) ⊂ 𝐻 𝑆 1 , 𝑛 0 , 0 ( 1 , 𝑝 ) , ( 𝛼 , 𝑝 ) ∈ 𝑃 , 𝐻 𝐶 1 , 𝑛 0 , 0 ( 0 , 𝑝 ) ⊂ 𝐻 𝐶 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) ⊂ 𝐻 𝐶 1 , 𝑛 0 , 0 ( 1 , 𝑝 ) , ( 𝛼 , 𝑝 ) ∈ 𝑃 . ( 1 . 7 )

2. Main Result

Directly from the definition of the class 𝐻 𝑆 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) ( 𝐻 𝐶 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) ) we get the following.

Theorem 2.1. Let ( 𝛼 , 𝑝 ) ∈ 𝑃 . If 𝑓 ∈ 𝐻 𝑆 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) ( 𝐻 𝐶 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) ) , then functions 𝑧 ⟼ 𝑟 − 1 𝑓 ( 𝑟 𝑧 ) , 𝑧 ⟼ 𝑒 − 𝑖 𝑡 𝑓 𝑒 𝑖 𝑡 𝑧 , 𝑧 ∈ 𝐸 , 𝑟 ∈ ( 0 , 1 ) , 𝑡 ∈ 𝑅 ( 2 . 1 ) also belong to 𝐻 𝑆 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) ( 𝐻 𝐶 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) ) .

Theorem 2.2. If 0 ≤ 𝛼 1 ≤ 𝛼 2 ≤ 1 ,   𝑝 > 0 , then 𝐻 𝑆 𝜆 𝑚 , 𝑛 1 , 𝜆 2 𝛼 1 , 𝑝 ⊂ 𝐻 𝑆 𝜆 𝑚 , 𝑛 1 , 𝜆 2 𝛼 2 , 𝑝 , 𝐻 𝐶 𝜆 𝑚 , 𝑛 1 , 𝜆 2 𝛼 1 , 𝑝 ⊂ 𝐻 𝐶 𝜆 𝑚 , 𝑛 1 , 𝜆 2 𝛼 2 . , 𝑝 ( 2 . 2 ) If 𝛼 ∈ [ 0 , 1 ] and 0 < 𝑝 1 ≤ 𝑝 2 , then 𝐻 𝑆 𝜆 𝑚 , 𝑛 1 , 𝜆 2 𝛼 , 𝑝 1 ⊃ 𝐻 𝑆 𝜆 𝑚 , 𝑛 1 , 𝜆 2 𝛼 , 𝑝 2 , 𝐻 𝐶 𝜆 𝑚 , 𝑛 1 , 𝜆 2 𝛼 , 𝑝 1 ⊃ 𝐻 𝐶 𝜆 𝑚 , 𝑛 1 , 𝜆 2 𝛼 , 𝑝 2 . ( 2 . 3 )

Theorem 2.3. Let ( 𝛼 , 𝑝 ) ∈ 𝑝 . If 𝑝 ≥ 1 , then every function 𝑓 ∈ 𝐻 𝐶 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 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 ∈ 𝐻 𝐶 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) is univalent and maps the unit disk 𝐸 onto a convex domain.

Proof. If 𝑝 ≥ 1 , then 𝑈 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) ≥ 𝑛 for 𝑛 = 2 , 3 , … , 𝛼 ∈ [ 0 , 1 ] , so by the condition (1.5) we obtain ∞ 𝑛 = 2 𝑛 | | 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 = 2 𝑛 2 | | 𝑎 𝑛 | | + | | 𝑏 𝑛 | | ≤ 1 . ( 2 . 5 ) Hence (see [2]), 𝑓 is convex.

Next, let 𝛼 ∈ [ 0 , 1 ] and set 𝑝 1 ( 𝛼 ) = 1 − l o g 2 ( 2 − 𝛼 ) , 𝑝 2 ( 𝛼 ) = 2 − l o g 2 ( 2 − 𝛼 ) ,   l o g 2 1 = 0 . We denote 𝐷 1 = ( 𝛼 , 𝑝 ) ∈ 𝑃 ∶ 𝑝 ≥ 𝑝 1 , 𝐷 ( 𝛼 ) 2 = ( 𝛼 , 𝑝 ) ∈ 𝑃 ∶ 𝑝 ≥ 𝑝 2 ( . 𝛼 ) ( 2 . 6 ) The next theorem present results concerning starlikeness and convexity of functions of the class 𝐻 𝐶 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) for arbitrary ( 𝛼 , 𝑝 ) ∈ 𝐷 1 and ( 𝛼 , 𝑝 ) ∈ 𝐷 2 , respectively.

Theorem 2.4. If ( 𝛼 , 𝑝 ) ∈ 𝐷 1 , then the functions of the class 𝐻 𝐶 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) are starlike.

Proof. We can check that the following inequality: 𝑈 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) ≥ 𝑛 , ( 𝛼 , 𝑝 ) ∈ 𝐷 1 , 𝑛 = 2 , 3 , … , ( 2 . 7 ) hold. If 𝑓 ∈ 𝐻 𝐶 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) for ( 𝛼 , 𝑝 ) ∈ 𝐷 1 , 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 ⧵ 𝐷 1 . If 𝑟 ∈ ( 0 , 𝑟 0 ( 𝛼 , 𝑝 ) ) , where 𝑟 0 ( 𝛼 , 𝑝 ) = 2 𝑝 − 1 ( 2 − 𝛼 ) , then each function 𝑓 ∈ 𝐻 𝐶 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 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   𝐸 1 = 𝐸 .

Proof. For ( 𝛼 , 𝑝 ) ∈ 𝑝 ⧵ 𝐷 1 , we have 𝑟 0 ( 𝛼 , 𝑝 ) < 1 , let 𝑓 ∈ 𝐻 𝐶 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) , ( 𝛼 , 𝑝 ) ∈ 𝑝 ⧵ 𝐷 1 , and let 𝑟 ∈ ( 0 , 𝑟 0 ( 𝛼 , 𝑝 ) ) . By Theorem 2.1, the function 𝑓 𝑟 of the form 𝑓 𝑟 ( 𝑧 ) = 𝑟 − 1 𝑓 ( 𝑟 𝑧 ) belongs to the class 𝐻 𝐶 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) and we have ∞ 𝑛 = 2 𝑛 | | 𝑎 𝑛 𝑟 𝑛 − 1 | | + | | 𝑏 𝑛 𝑟 𝑛 − 1 | | = ∞ 𝑛 = 2 𝑛 𝑟 𝑛 − 1 | | 𝑎 𝑛 | | + | | 𝑏 𝑛 | | . ( 2 . 8 ) In view of properties of elementary functions, we obtain 𝑛 𝑟 𝑛 − 1 𝑟 ≤ 𝑛 0 ( 𝛼 , 𝑝 ) 𝑛 − 1 ≤ 𝑈 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) , 𝑛 = 2 , 3 , … . ( 2 . 9 ) Hence, 𝑓 𝑟 ∈ 𝐻 𝑆 1 , 𝑛 0 , 0 ( 1 , 1 ) [2] for any 𝑟 ∈ ( 0 , 𝑟 0 ( 𝛼 , 𝑝 ) ) maps the 𝐸 onto a domain starlike with respect to the origin.

Theorem 2.6. Let ( 𝛼 , 𝑝 ) ∈ 𝑝 ⧵ 𝐷 2 . If 𝑟 ∈ ( 0 , 𝑟 ∗ 0 ( 𝛼 , 𝑝 ) ) , where 𝑟 ∗ 0 ( 𝛼 , 𝑝 ) = 2 𝑝 − 2 ( 2 − 𝛼 ) , then each function 𝑓 ∈ 𝐻 𝐶 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) maps the disk 𝐸 𝑟 onto a convex domain.

Proof. For every ( 𝛼 , 𝑝 ) ∈ 𝑝 ⧵ 𝐷 2 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 ) ) 𝑛 2 𝑟 𝑛 − 1 ≤ 𝑈 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) , 𝑛 = 2 , 3 , … . ( 2 . 1 0 ) Hence 𝑓 𝑟 ∈ 𝐻 𝐶 1 , 𝑛 0 , 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 ∈ 𝐻 𝑆 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) , 𝑧 ∈ 𝐸 , 𝑧 ≠ 0 , then | | | | ≤ | | 𝑏 𝑓 ( 𝑧 ) 1 + 1 | | | | 𝑏 | 𝑧 | + 1 − 1 | | 2 𝑝 ( 2 − 𝛼 ) | 𝑧 | 2 , | | | | ≥ | | 𝑏 𝑓 ( 𝑧 ) 1 − 1 | | | | 𝑏 | 𝑧 | − 1 − 1 | | 2 𝑝 ( 2 − 𝛼 ) | 𝑧 | 2 . ( 2 . 1 1 )

Proof. Let 𝑓 ∈ 𝐻 𝑆 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 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 | | ≤ | | 𝑏 1 − 1 | | 𝑈 𝜆 𝑚 , 2 1 , 𝜆 2 ( − 𝛼 , 𝑝 ) ∞ 𝑛 = 3 ⎛ ⎜ ⎜ ⎝ 𝑈 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) 𝑈 𝜆 𝑚 , 2 1 , 𝜆 2 ( ⎞ ⎟ ⎟ ⎠ | | 𝑎 𝛼 , 𝑝 ) − 1 𝑛 | | + | | 𝑏 𝑛 | | . ( 2 . 1 2 ) Since 𝑈 𝜆 𝑚 , 𝑛 1 , 𝜆 2 ( 𝛼 , 𝑝 ) ≥ 𝑈 𝜆 𝑚 , 2 1 , 𝜆 2 ( 𝛼 , 𝑝 ) , 𝑛 = 3 , 4 , … , ( 𝛼 , 𝑝 ) ∈ 𝑃 , we have ∞ 𝑛 = 2 | | 𝑎 𝑛 | | + | | 𝑏 𝑛 | | ≤ | | 𝑏 1 − 1 | | 𝑈 𝜆 𝑚 , 2 1 , 𝜆 2 . ( 𝛼 , 𝑝 ) ( 2 . 1 3 ) Hence, | | | | ≤ 𝑓 ( 𝑧 ) ∞ 𝑛 = 2 | | 𝑎 𝑛 | | + | | 𝑏 𝑛 | | | 𝑧 | 𝑛 + | | 𝑏 1 + 1 | | | | | 𝑏 𝑧 | ≤ 1 + 1 | | | | | 𝑏 𝑧 | + 1 − 1 | | 𝑈 𝜆 𝑚 , 2 1 , 𝜆 2 | ( 𝛼 , 𝑝 ) 𝑧 | 2 , ( 2 . 1 4 ) that is, the upper estimate.
The lower estimate follows from (2.13) and the inequality: | | | | | | 𝑏 𝑓 ( 𝑧 ) ≥ | 𝑧 | − 1 | | | 𝑧 | − ∞ 𝑛 = 2 | | 𝑎 𝑛 | | + | | 𝑏 𝑛 | | | 𝑧 | 𝑛 . ( 2 . 1 5 )

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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