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.
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.
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  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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