Open Access

On the convex-exponent product of logharmonic mappings

Journal of Inequalities and Applications20142014:485

https://doi.org/10.1186/1029-242X-2014-485

Received: 18 April 2014

Accepted: 19 November 2014

Published: 8 December 2014

Abstract

Sufficient conditions are obtained on two given logharmonic mappings f 1 and f 2 that ensure the product F ( z ) = f 1 λ ( z ) f 2 1 λ ( z ) , 0 λ 1 , is a univalent starlike logharmonic mapping. Several illustrative examples are constructed from this product.

MSC: 30C35, 30C45, 35Q30.

Keywords

logharmonic mappings starlike mappings spirallike mappings

1 Introduction

Let U = { z : | z | < 1 } be the unit disk in the complex plane , and let B denote the set of bounded analytic functions a satisfying | a ( z ) | < 1 in U. Also let B 0 denote its subclass consisting of a B with a ( 0 ) = 0 . A logharmonic mapping f defined in U is a solution of the nonlinear elliptic partial differential equation
f z ¯ ¯ f ¯ = a f z f ,
where the second dilatation function a lies in B. Thus the Jacobian
J f = | f z | 2 ( 1 | a | 2 )
is positive, and all non-constant logharmonic mappings are therefore sense-preserving and open in U. If f is a non-constant logharmonic mapping that vanishes only at z = 0 , then f admits the representation
f ( z ) = z m | z | 2 β m h ( z ) g ( z ) ¯ ,
(1)
where m is a nonnegative integer, Re β > 1 / 2 , while h and g are analytic functions in U satisfying g ( 0 ) = 1 and h ( 0 ) 0 (see [1]). The exponent β in (1) depends only on a ( 0 ) and is given by
β = a ( 0 ) ¯ 1 + a ( 0 ) 1 | a ( 0 ) | 2 .
Note that f ( 0 ) 0 if and only if m = 0 , and that a univalent logharmonic mapping in U vanishes at the origin if and only if m = 1 , that is, f has the form
f ( z ) = z | z | 2 β h ( z ) g ( z ) ¯ ,

where Re β > 1 / 2 and 0 ( h g ) ( U ) . This class has been studied extensively in recent years, for instance, in [18], and [9].

In this case, F ( ζ ) = log f ( e ζ ) is a univalent harmonic mapping of the half-plane { ζ : Re ( ζ ) < 0 } . Studies on univalent harmonic mappings can be found in [1016], and [17]. Such mappings are closely related to the theory of minimal surfaces [18, 19].

In this work, emphasis is given on univalent and sense-preserving logharmonic mappings in U with respect to a B 0 . These mappings are of the form
f ( z ) = z h ( z ) g ( z ) ¯ ,
(2)

and the class consisting of such mappings is denoted by S L h . Also let S L h denote its subclass of univalent starlike logharmonic mappings. The classical family S of univalent analytic starlike functions is evidently a subclass of S L h . The representation in (2) is essential to the present work as it allows the treatment of logharmonic mappings f through their associated analytic representations h and g (see [35], and [6]). For example, Abdulhadi and Abu Muhanna [5] established a connection between starlike logharmonic mappings of order α and starlike analytic functions of order α.

It follows from (2) that the functions h, g and the dilatation a satisfy
z g ( z ) g ( z ) = a ( z ) ( 1 + z h ( z ) h ( z ) ) .
(3)
Given an analytic function φ with a specified geometric property and a B 0 , a common method to construct a logharmonic mapping f ( z ) = z h ( z ) g ( z ) ¯ is to solve for h and g via the equations
z h ( z ) g ( z ) = φ ( z ) and z g ( z ) g ( z ) 1 + z h ( z ) h ( z ) = a ( z ) .
Thus the solution is f ( z ) = z h ( z ) g ( z ) ¯ with
g ( z ) = exp 0 z a ( s ) 1 a ( s ) φ ( s ) φ ( s ) d s and h ( z ) = φ ( z ) g ( z ) z .

In this paper, a new logharmonic mapping with a specified property is constructed by taking product combination of two functions possessing the given property. Specifically, if f 1 ( z ) = z h 1 ( z ) g 1 ( z ) ¯ with respect to a 1 B 0 , and f 2 ( z ) = z h 2 ( z ) g 2 ( z ) ¯ with respect to a 2 B 0 , we construct a new univalent logharmonic mapping F ( z ) = f 1 λ ( z ) f 2 1 λ ( z ) , 0 λ 1 , with respect to μ B 0 . Sufficient conditions are obtained on f 1 and f 2 for the product combination F ( z ) = f 1 λ ( z ) f 2 1 λ ( z ) to be starlike. We close the work by giving several examples of univalent starlike logharmonic mappings constructed from this product.

2 Product of logharmonic mappings

Let Ω be a simply connected domain in containing the origin. Then Ω is said to be α-spirallike, π / 2 < α < π / 2 , if w exp ( t e i α ) Ω for all t 0 whenever w Ω . Evidently, Ω is starlike (with respect to the origin) if α = 0 .

The following result from [6] will be needed in the sequel.

Lemma 1 Let f ( z ) = z h ( z ) g ( z ) ¯ be logharmonic in U with 0 h g ( U ) . Then f S L h if and only if φ ( z ) = z h ( z ) / g ( z ) S .

Theorem 1 Let f ( z ) = z h ( z ) g ( z ) ¯ S L h with respect to a B 0 , and let γ be a constant with Re γ > 1 / 2 . Then F ( z ) = f ( z ) | f ( z ) | 2 γ is an α-spirallike logharmonic mapping with respect to
a ˆ ( z ) = 1 + γ ¯ 1 + γ a ( z ) + γ ¯ 1 + γ ¯ 1 + a ( z ) γ 1 + γ = ( 1 + γ ¯ ) a ( z ) + γ ¯ 1 + γ + γ a ( z ) ,

where α = tan 1 ( 2 Im γ / ( 1 + 2 Re γ ) ) .

Proof The function F = f | f | 2 γ = f 1 + γ f ¯ γ is logharmonic with respect to a ˆ = ( F z ¯ ¯ / F ¯ ) / ( F z / F ) . Indeed,
a ˆ ( z ) = ( 1 + γ ¯ ) a ( z ) f z f + γ ¯ ( f ¯ z ¯ f ¯ ) ¯ ( 1 + γ ) f z f + γ f z ¯ ¯ f ¯ = ( 1 + γ ¯ ) a ( z ) f z f + γ ¯ f z f ( 1 + γ ) f z f + γ a ( z ) f z f = 1 + γ ¯ 1 + γ a ( z ) + γ ¯ 1 + γ ¯ 1 + a ( z ) γ 1 + γ .
Thus
| a ˆ ( z ) | = | a ( z ) + γ ¯ 1 + γ ¯ 1 + a ( z ) γ 1 + γ | < 1

provided | γ | 2 < | 1 + γ | 2 , which evidently holds since Re γ > 1 / 2 .

Also F = f | f | 2 γ = f 1 + γ f ¯ γ = z | z | 2 γ h 1 + γ g γ h γ ¯ g 1 + γ ¯ ¯ . Let H = h 1 + γ g γ , G = h γ ¯ g 1 + γ ¯ , and ψ ( z ) = z H ( z ) / G ( z ) e 2 i α . Then
e i α z ψ ( z ) ψ ( z ) = e i α + ( ( 1 + γ ) e i α γ ¯ e i α ) z h ( z ) h ( z ) ( ( 1 + γ ¯ ) e i α γ e i α ) z g ( z ) g ( z ) .
The condition on α ensures that
( 1 + γ ) e i α γ ¯ e i α cos α = ( 1 + γ ¯ ) e i α γ e i α cos α = 1 .
Also Lemma 1 shows that φ ( z ) = z h ( z ) / g ( z ) S . Thus
Re ( e i α z ψ ( z ) ψ ( z ) ) = ( cos α ) Re ( z φ ( z ) φ ( z ) ) > 0 ,

and it follows from [[6], Theorem 2.1] that F is α-spirallike logharmonic whose dilatation is a ˆ ( z ) . □

Remark 1 Observe that F in Theorem 1 is starlike if and only if γ > 1 / 2 .

Theorem 2 Let f k ( z ) = z h k ( z ) g k ( z ) ¯ S L h ( k = 1 , 2 ) with respect to the same a B 0 . Then F ( z ) = f 1 λ ( z ) f 2 1 λ ( z ) , 0 λ 1 , is a univalent starlike logharmonic mapping with respect to the same a.

Proof Let μ = ( F z ¯ ¯ / F ¯ ) / ( F z / F ) . It follows from (3) that
μ = λ f 1 z ¯ ¯ f 1 ¯ + ( 1 λ ) f 2 z ¯ ¯ f 2 ¯ λ f 1 z f 1 + ( 1 λ ) f 2 z f 2 = λ g 1 g 1 + ( 1 λ ) g 2 g 2 λ ( z h 1 ) ( z h 1 ) + ( 1 λ ) ( z h 2 ) ( z h 2 ) = λ a ( z h 1 ) ( z h 1 ) + ( 1 λ ) a ( z h 2 ) ( z h 2 ) λ ( z h 1 ) ( z h 1 ) + ( 1 λ ) ( z h 2 ) ( z h 2 ) = a .
(4)

Hence | μ ( z ) | < 1 in U, which implies that F is a locally univalent logharmonic mapping.

Next F is shown to have the form (2). Since f 1 = z h 1 g 1 ¯ , and f 2 = z h 2 g 2 ¯ , then
F ( z ) = f 1 λ ( z ) f 2 1 λ ( z ) = ( z h 1 ( z ) g 1 ( z ) ¯ ) λ ( z h 2 ( z ) g 2 ( z ) ¯ ) 1 λ = z h 1 λ h 2 1 λ g 1 λ ( z ) g 2 1 λ ( z ) ¯ = z h ( z ) g ( z ) ¯
(5)

with h = h 1 λ h 2 1 λ and g = g 1 λ g 2 1 λ .

Since f k is starlike, that is, each φ k = z h k / g k satisfies the condition Re z φ k ( z ) / φ k ( z ) > 0 in U, direct computations show that
arg ( F ( r e i θ ) ) θ = Re ( z F z F z ¯ F z ¯ F ) = λ Re ( z f 1 z f 1 z ¯ f 1 z ¯ f 1 ) + ( 1 λ ) Re ( z f 2 z f 2 z ¯ f 2 z ¯ f 2 ) = λ Re ( z φ 1 ( z ) φ 1 ( z ) ) + ( 1 λ ) Re ( z φ 2 ( z ) φ 2 ( z ) ) > 0 .

Thus F is starlike. □

The following corollary is an immediate consequence of Theorem 2.

Corollary 1 Let f k ( z ) = z h k ( z ) g k ( z ) ¯ S L h ( k = 1 , 2 , , n ) with respect to the same a B 0 . Then F = f 1 λ 1 f 2 λ 2 f n λ n is a univalent starlike logharmonic mapping with respect to the same a, where 0 λ k 1 and λ 1 + λ 2 + + λ n = 1 .

Theorem 3 Let f k ( z ) = z h k ( z ) g k ( z ) ¯ S L h ( k = 1 , 2 ) with respect to a k B 0 . Suppose also that
Re ( 1 a 1 a 2 ¯ ) ( z h 1 ) ( z h 1 ) ( ( z h 2 ) ( z h 2 ) ) ¯ 0 .

Then F ( z ) = f 1 λ ( z ) f 2 1 λ ( z ) , 0 λ 1 , is a univalent starlike logharmonic mapping.

Proof The argument is similar to the proof of Theorem 2. From (5), evidently F has the form (2).

Let ( F z / F ) μ ( z ) = F z ¯ ¯ / F ¯ . Since | a k | < 1 , it follows from (3) and (4) that
| μ ( z ) | = | λ g 1 g 1 + ( 1 λ ) g 2 g 2 λ ( z h 1 ) ( z h 1 ) + ( 1 λ ) ( z h 2 ) ( z h 2 ) | = | λ a 1 ( z h 1 ) ( z h 1 ) + ( 1 λ ) a 2 ( z h 2 ) ( z h 2 ) λ ( z h 1 ) ( z h 1 ) + ( 1 λ ) ( z h 2 ) ( z h 2 ) | .
(6)
By assumption,
| λ ( z h 1 ) ( z h 1 ) + ( 1 λ ) ( z h 2 ) ( z h 2 ) | 2 | λ a 1 ( z h 1 ) ( z h 1 ) + ( 1 λ ) a 2 ( z h 2 ) ( z h 2 ) | 2 = λ 2 ( 1 | a 1 | 2 ) | ( z h 1 ) ( z h 1 ) | 2 + ( 1 λ ) 2 ( 1 | a 2 | 2 ) | ( z h 2 ) ( z h 2 ) | 2 + 2 λ ( 1 λ ) Re ( ( 1 a 1 a 2 ¯ ) ( z h 1 ) ( z h 1 ) ( ( z h 2 ) ( z h 2 ) ) ¯ ) > 0 ,

whence | μ ( z ) | < 1 , which implies that F is locally univalent.

Now the associated analytic function for F is given by φ = ( z h 1 λ h 2 1 λ ) / ( g 1 λ g 2 1 λ ) . Let φ ( U ) = Ω . From Lemma 1, φ k = z h k / g k S , and thus
Re ( z φ ( z ) φ ( z ) ) = λ Re ( z φ 1 ( z ) φ 1 ( z ) ) + ( 1 λ ) Re ( z φ 2 ( z ) φ 2 ( z ) ) > 0 .

Hence Ω is a starlike domain, and we deduce that F is a univalent starlike logharmonic mapping. □

Theorem 4 Let f k = z h k g k ¯ S L h with respect to a k B , k = 1 , 2 , satisfying z h k g k = z . Then F ( z ) = f 1 λ ( z ) f 2 1 λ ( z ) , 0 λ 1 , is a univalent starlike logharmonic mapping.

Proof Since
( z h k ) ( z h k ) + g k g k = 1 z ,
it follows from (3) that
( z h k ) ( z h k ) = 1 z ( 1 + a k ) .
(7)
With F ( z ) = f 1 λ ( z ) f 2 1 λ ( z ) , (6) and (7) readily yield
| μ ( z ) | = | λ a 1 + ( 1 λ ) a 2 + a 1 a 2 1 + ( 1 λ ) a 1 + λ a 2 | .

Evidently | μ ( z ) | < 1 is equivalent to ψ ( λ ) = | 1 + ( 1 λ ) a 1 + λ a 2 | 2 | λ a 1 + ( 1 λ ) a 2 + a 1 a 2 | 2 > 0 .

Now
ψ ( λ ) = 2 λ ( ( 1 | a 1 | 2 ) Re a 2 ( 1 | a 2 | 2 ) Re a 1 ( | a 1 | 2 | a 2 | 2 ) ) + ( 1 | a 2 | 2 ) | 1 + a 1 | 2
is a continuous monotonic function of λ in the interval [ 0 , 1 ] . Since
ψ ( 0 ) = ( 1 | a 2 | 2 ) | 1 + a 1 | 2 > 0
and
ψ ( 1 ) = ( 1 | a 1 | 2 ) | 1 + a 2 | 2 > 0 ,

we deduce that ψ ( λ ) > 0 for all λ [ 0 , 1 ] , and thus F is locally univalent.

With φ k = z h k / g k , then
Re ( z φ k ( z ) φ k ( z ) ) = Re ( ( 1 a k ) z ( z h k ) ( z h k ) ) = Re ( 1 a k 1 + a k ) > 0 .

Hence φ k is starlike univalent, and from Lemma 1, f k ( z ) = z h k ( z ) g k ( z ) ¯ is starlike univalent logharmonic.

The associated analytic function for F is given by φ ( z ) = ( z h 1 λ h 2 1 λ ) / ( g 1 λ g 2 1 λ ) . Further
Re ( z φ ( z ) φ ( z ) ) = λ Re ( z φ 1 ( z ) φ 1 ( z ) ) + ( 1 λ ) Re ( z φ 2 ( z ) φ 2 ( z ) ) > 0 ,

and thus F is starlike. □

The proof of Theorem 4 evidently gives the following result of [[6], Lemma 3.1 and Theorem 3.2].

Corollary 2 Let f k = z h k g k ¯ S L h ( k = 1 , 2 ) with respect to a k B 0 , and suppose that z h k g k = z . Then φ ( z ) = z ( h k ( z ) ) 2 S .

3 Examples

We give several illustrative examples in this section.

Example 1 Let
f ( z ) = z ( 1 z ¯ 1 z ) .

Then f is a univalent logharmonic mapping with respect to a ( z ) = z , and it maps U onto U [6].

Now the function F ( z ) = f ( z ) | f ( z ) | 2 γ is an α-spirallike logharmonic mapping with respect to
a ˆ ( z ) = 1 + γ ¯ 1 + γ z + γ ¯ 1 + γ ¯ 1 z γ 1 + γ ,
where α = tan 1 ( 2 Im γ / ( 1 + 2 Re γ ) ) . In particular, if γ = i , then α = tan 1 ( 2 ) = 0.352 π , and
a ˆ ( z ) = i ( 1 i ) z 1 + i i z .
The image of circles in the unit disk under f is shown in Figure 1, and Figure 2 shows the image of the radial slits in U by F.
Figure 1

Graph of circles in U by f ( z ) = z ( 1 z ¯ ) 1 z .

Figure 2

Graph of radial slits by F ( z ) = f ( z ) | f ( z ) | 2 i , f ( z ) = z ( 1 z ¯ ) 1 z .

Example 2 Consider the functions
f 1 ( z ) = z ( 1 z ¯ 1 z ) exp { Re 4 z 1 z } and f 2 ( z ) = z ( 1 + z ¯ 1 + z ) .

Since φ 1 ( z ) = z / ( 1 z ) 2 and φ 2 ( z ) = z / ( 1 + z ) 2 are starlike analytic functions, it follows from [[5], Theorem 1] that f 1 and f 2 are starlike logharmonic mappings with respect to a ( z ) = z . Theorem 2 shows that F ( z ) = f 1 λ ( z ) f 2 1 λ ( z ) , 0 λ 1 , is a starlike univalent logharmonic mapping.

The image of F is shown in Figure 3 for λ = 1 / 3 .
Figure 3

Graph of circles in U by F ( z ) = z ( 1 z ¯ 1 z exp { Re 4 z 1 z } ) 1 / 3 ( 1 + z ¯ 1 + z ) 2 / 3 .

Example 3 In this example, let
f 1 ( z ) = z ( 1 z ¯ ) ( 1 z ) and f 2 ( z ) = z ( 1 z ¯ ) ( 1 z ) exp { Re 4 z 1 z } ,
and
F ( z ) = f 1 λ ( z ) f 2 1 λ ( z ) , 0 λ 1 .

Simple calculations show that f 1 and f 2 are respectively starlike logharmonic with dilatations a 1 ( z ) = z and a 2 ( z ) = z . Also F is logharmonic with respect to μ ( z ) = z ( ( 1 2 λ ) + z ) / ( 1 + ( 1 2 λ ) z ) .

Since
Re ( ( 1 a 1 a 2 ¯ ) ( z h 1 ) ( z h 1 ) ( ( z h 2 ) ( z h 2 ) ) ¯ ) = Re ( ( 1 + | z | 2 ) 1 z ( 1 z ) 1 + z ¯ z ¯ ( 1 z ¯ ) 2 ) = ( 1 + | z | 2 ) | z | 2 | 1 z | 2 Re 1 + z 1 z > 0 ,

the conditions of Theorem 3 are satisfied, and thus F is starlike univalent.

The image of circles in U under F for λ = 1 / 3 is shown in Figure 4.
Figure 4

Graph of F ( z ) = z ( 1 z ¯ 1 z ) 1 / 3 ( 1 z ¯ 1 z exp { Re 4 z 1 z } ) 2 / 3 .

Example 4 Let f 1 ( z ) = z h 1 ( z ) g 1 ( z ) ¯ , where z h 1 ( z ) g 1 ( z ) = z , a 1 ( z ) = z , and
h 1 ( z ) = 1 1 + z , g 1 ( z ) = 1 + z .
Thus
f 1 ( z ) = z ( 1 + z ¯ ) ( 1 + z ) .
Further, let f 2 ( z ) = z h 2 ( z ) g 2 ( z ) ¯ , where z h 2 ( z ) g 2 ( z ) = z , a 2 ( z ) = z 2 , and
h 2 ( z ) = 1 1 + z 2 , g 2 ( z ) = 1 + z 2 .
In this case,
f 2 ( z ) = z 1 + z ¯ 2 1 + z 2 .
Since f 1 and f 2 satisfy the conditions of Theorem 4, we deduce that F ( z ) = f 1 λ ( z ) f 2 1 λ ( z ) , 0 λ 1 , is a univalent starlike logharmonic mapping. The image of U under F for λ = 1 / 3 is shown in Figure 5.
Figure 5

Graph of F ( z ) = z ( 1 + z ¯ 1 + z ) 1 / 3 ( 1 + z ¯ 2 1 + z 2 ) 2 / 3 .

Declarations

Acknowledgements

This work was completed when the first author was visiting Universiti Sains Malaysia (USM). The work presented here was supported in parts by the FRGS and USM-RU research grants. The authors are thankful to the referees for the suggestions that helped improve the clarity of this manuscript.

Authors’ Affiliations

(1)
Department of Mathematics, American University of Sharjah
(2)
Department of Mathematics, University of Dammam
(3)
School of Mathematical Sciences, Universiti Sains Malaysia (USM)

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