- Open Access
On the convex-exponent product of logharmonic mappings
© AbdulHadi et al.; licensee Springer. 2014
- Received: 18 April 2014
- Accepted: 19 November 2014
- Published: 8 December 2014
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.
- logharmonic mappings
- starlike mappings
- spirallike mappings
In this case, is a univalent harmonic mapping of the half-plane . Studies on univalent harmonic mappings can be found in [10–16], and . Such mappings are closely related to the theory of minimal surfaces [18, 19].
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 ). For example, Abdulhadi and Abu Muhanna  established a connection between starlike logharmonic mappings of order α and starlike analytic functions of order α.
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.
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  will be needed in the sequel.
Lemma 1 Let be logharmonic in U with . Then if and only if .
provided , which evidently holds since .
and it follows from [, 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.
Hence in U, which implies that F is a locally univalent logharmonic mapping.
with and .
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 .
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).
whence , which implies that F is locally univalent.
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.
Evidently is equivalent to .
we deduce that for all , and thus F is locally univalent.
Hence is starlike univalent, and from Lemma 1, is starlike univalent logharmonic.
and thus F is starlike. □
The proof of Theorem 4 evidently gives the following result of [, Lemma 3.1 and Theorem 3.2].
Corollary 2 Let () with respect to , and suppose that . Then .
We give several illustrative examples in this section.
Then f is a univalent logharmonic mapping with respect to , and it maps U onto U .
Since and are starlike analytic functions, it follows from [, Theorem 1] that and are starlike logharmonic mappings with respect to . Theorem 2 shows that , , is a starlike univalent logharmonic mapping.
Simple calculations show that and are respectively starlike logharmonic with dilatations and . Also F is logharmonic with respect to .
the conditions of Theorem 3 are satisfied, and thus F is starlike univalent.
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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