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Estimating coefficients for certain subclasses of meromorphic and bi-univalent functions
- F. Müge Sakar^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s13660-018-1879-4
© The Author(s) 2018
- Received: 13 April 2018
- Accepted: 10 October 2018
- Published: 19 October 2018
Abstract
In the present paper, we introduce two interesting subclasses of meromorphic and bi-univalent functions defined on \(\Delta=\{z:z\in\mathbb{C}, 1<|z|<\infty\}\). For functions belonging to these subclasses, estimates on the initial coefficient \(|b_{0}|\) and \(|b_{1}|\) are obtained. Some other closely related results are also represented. The coefficient bounds presented here are new in their own kind. We hope that this paper will generate further interest in applying our approach to other related problems.
Keywords
- Univalent function
- Bi-univalent function
- Meromorphic function
- Taylor–Maclaurin coefficients
MSC
- 30C45
- 30C50
1 Introduction and lemmas
A function \(f\in\mathcal{A}\) is said to be bi-univalent in the open unit disk \(\mathbb{U}\) if both the function and its inverse are univalent in \(\mathbb{U}\). Let σ denote the class of analytic and bi-univalent functions in \(\mathbb{U}\) given by the Taylor–Maclaurin series expansion as in (1.1). For a brief history and interesting examples of functions in the class σ, see [19]. In fact, the aforecited work of Srivastava et al. [19] essentially revived the investigation of various subclasses of the bi-univalent function class σ in recent years; it was followed by works of, e.g., Frasin and Aouf [6], Srivastava et al. [18, 20], Xu et al. [21, 22] and others (see, for example, [1, 2, 4, 7, 11, 14]).
In 1977, Kubota [9] proved that the Springer conjecture is true for \(n=3,4,5\), and subsequently Schober [16] obtained sharp bounds for the coefficients \(B_{2n-1}\) (\(1\leq n \leq7\)). Recently, Kapoor and Mishra [8] found coefficient estimates for inverses of meromorphic starlike functions of positive order α in Δ.
In the present investigation, certain subclasses of meromorphic bi-univalent functions are introduced and estimates for the coefficients \(|b_{0}|\) and \(|b_{1}|\) of functions in the newly introduced subclasses are obtained. These coefficient results are obtained by associating with the functions having positive real part. An analytic function p of the form \(p(z)=1+c_{1}z+c_{2}z^{2}+\cdots\) is called a function with positive real part in \(\mathbb{U}\) if \(\Re p(z)>0\) for all \(z\in\mathbb{U}\). The class of all functions with positive real part is denoted by \(\mathcal{P}\). We need the following lemmas [13] to prove our main results.
Lemma 1.1
In 1972, the following univalence criterion was proved by Ozaki and Nunokawa [12].
Lemma 1.2
2 Main results
Definition 2.1
Definition 2.2
Theorem 2.1
Proof
Theorem 2.2
Proof
3 Conclusion
Lemma 3.1
Example 3.2
Corollary 3.1
Proof
Assume that the function \(g(z)=z+\sum_{n=1}^{\infty}\frac{b_{n}}{z^{n}}\in\mathcal{T}_{\Sigma_{\sigma}}(\mu)\) where \(0<\mu\leq1\). Since \(b_{0}=0, \ s_{1}=t_{1}=0\), the result can be verified by a direct calculation of (2.36). □
Corollary 3.2
Proof
Since the function \(g(z)=z+\sum_{n=1}^{\infty}\frac{b_{n}}{z^{n}}\in\mathcal{T}_{\Sigma_{\sigma}}^{\alpha}\) where \(0<\alpha\leq1\) and \(b_{0}=0\), it follows that \(s_{1}=t_{1}=0\). The result can now be seen by a direct calculation of (2.17). □
Declarations
Acknowledgements
The author is indebted to Professor Abdallah Lyzzaik and Professor Daoud Bshouty for their valuable contributions to this paper.
Funding
The present investigation was partly supported by Batman University Scientific Research Project Coordination Unit. Project Number: BTUBAP-2018-IIBF-2.
Authors’ contributions
Author read and approved the final manuscript.
Competing interests
The author declares that she has no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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