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On the convex-exponent product of logharmonic mappings
Journal of Inequalities and Applications volume 2014, Article number: 485 (2014)
Abstract
Sufficient conditions are obtained on two given logharmonic mappings and that ensure the product , , is a univalent starlike logharmonic mapping. Several illustrative examples are constructed from this product.
MSC: 30C35, 30C45, 35Q30.
1 Introduction
Let be the unit disk in the complex plane â„‚, and let B denote the set of bounded analytic functions a satisfying in U. Also let denote its subclass consisting of with . A logharmonic mapping f defined in U is a solution of the nonlinear elliptic partial differential equation
where the second dilatation function a lies in B. Thus the Jacobian
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 , then f admits the representation
where m is a nonnegative integer, , while h and g are analytic functions in U satisfying and (see [1]). The exponent β in (1) depends only on and is given by
Note that if and only if , and that a univalent logharmonic mapping in U vanishes at the origin if and only if , that is, f has the form
where and . This class has been studied extensively in recent years, for instance, in [1–8], and [9].
In this case, is a univalent harmonic mapping of the half-plane . Studies on univalent harmonic mappings can be found in [10–16], 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 . These mappings are of the form
and the class consisting of such mappings is denoted by . Also let denote its subclass of univalent starlike logharmonic mappings. The classical family of univalent analytic starlike functions is evidently a subclass of . 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 [3–5], 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
Given an analytic function φ with a specified geometric property and , a common method to construct a logharmonic mapping is to solve for h and g via the equations
Thus the solution is with
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 with respect to , and with respect to , we construct a new univalent logharmonic mapping , , with respect to . Sufficient conditions are obtained on and for the product combination 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 ℂ containing the origin. Then Ω is said to be α-spirallike, , if for all whenever . Evidently, Ω is starlike (with respect to the origin) if .
The following result from [6] will be needed in the sequel.
Lemma 1 Let be logharmonic in U with . Then if and only if .
Theorem 1 Let with respect to , and let γ be a constant with . Then is an α-spirallike logharmonic mapping with respect to
where .
Proof The function is logharmonic with respect to . Indeed,
Thus
provided , which evidently holds since .
Also . Let , , and . Then
The condition on α ensures that
Also Lemma 1 shows that . Thus
and it follows from [[6], Theorem 2.1] that F is α-spirallike logharmonic whose dilatation is . □
Remark 1 Observe that F in Theorem 1 is starlike if and only if .
Theorem 2 Let () with respect to the same . Then , , is a univalent starlike logharmonic mapping with respect to the same a.
Proof Let . It follows from (3) that
Hence in U, which implies that F is a locally univalent logharmonic mapping.
Next F is shown to have the form (2). Since , and , then
with and .
Since is starlike, that is, each satisfies the condition in U, direct computations show that
Thus F is starlike. □
The following corollary is an immediate consequence of Theorem 2.
Corollary 1 Let () with respect to the same . Then is a univalent starlike logharmonic mapping with respect to the same a, where and .
Theorem 3 Let () with respect to . Suppose also that
Then , , 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 . Since , it follows from (3) and (4) that
By assumption,
whence , which implies that F is locally univalent.
Now the associated analytic function for F is given by . Let . From Lemma 1, , and thus
Hence Ω is a starlike domain, and we deduce that F is a univalent starlike logharmonic mapping. □
Theorem 4 Let with respect to , , satisfying . Then , , is a univalent starlike logharmonic mapping.
Proof Since
it follows from (3) that
With , (6) and (7) readily yield
Evidently is equivalent to .
Now
is a continuous monotonic function of λ in the interval . Since
and
we deduce that for all , and thus F is locally univalent.
With , then
Hence is starlike univalent, and from Lemma 1, is starlike univalent logharmonic.
The associated analytic function for F is given by . Further
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 () with respect to , and suppose that . Then .
3 Examples
We give several illustrative examples in this section.
Example 1 Let
Then f is a univalent logharmonic mapping with respect to , and it maps U onto U [6].
Now the function is an α-spirallike logharmonic mapping with respect to
where . In particular, if , then , and
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.
Example 2 Consider the functions
Since and are starlike analytic functions, it follows from [[5], Theorem 1] that and are starlike logharmonic mappings with respect to . Theorem 2 shows that , , is a starlike univalent logharmonic mapping.
The image of F is shown in Figure 3 for .
Example 3 In this example, let
and
Simple calculations show that and are respectively starlike logharmonic with dilatations and . Also F is logharmonic with respect to .
Since
the conditions of Theorem 3 are satisfied, and thus F is starlike univalent.
The image of circles in U under F for is shown in Figure 4.
Example 4 Let , where , , and
Thus
Further, let , where , , and
In this case,
Since and satisfy the conditions of Theorem 4, we deduce that , , is a univalent starlike logharmonic mapping. The image of U under F for is shown in Figure 5.
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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.
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The study was conceived and planned by all authors. Every author participated in the discussions of tackling the problem and the directions of the proofs of the results. All authors read and approved the final manuscript.
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AbdulHadi, Z., Alareefi, N.M. & Ali, R.M. On the convex-exponent product of logharmonic mappings. J Inequal Appl 2014, 485 (2014). https://doi.org/10.1186/1029-242X-2014-485
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DOI: https://doi.org/10.1186/1029-242X-2014-485