On a boundary property of analytic functions
- Mamoru Nunokawa^{1} and
- Janusz Sokół^{2}Email author
https://doi.org/10.1186/s13660-017-1575-9
© The Author(s) 2017
Received: 10 July 2017
Accepted: 22 November 2017
Published: 28 November 2017
Abstract
Let f be an analytic function in the unit disc \(|z|<1\) on the complex plane \(\mathbb {C}\). This paper is devoted to obtaining the correspondence between \(f(z)\) and \(zf'(z)\) at the point w, \(0<|w|=R< 1\), such that \(|f(w)|=\min \{|f(z)|: f(z)\in\partial f(|z|\leq R) \}\). We present several applications of the main result. A part of them improve the previous results of this type.
Keywords
MSC
1 Introduction
Let \({\mathcal {H}}\) denote the class of analytic functions in the unit disc \(|z|<1\) on the complex plane \({\mathbb{C}}\). The following lemma is a particular case of the Julia-Wolf theorem. It is known as Jack’s lemma.
Lemma 1.1
([1])
Let \(\omega(z)\in{\mathcal {H}}\) with \(\omega(0)=0\). If for a certain \(z_{0}\), \(|z_{0}|<1\), we have \(|\omega(z)|\leq|\omega(z_{0})|\) for \(|z|\leq|z_{0}|\), then \(z_{0}\omega'(z_{0})=m\omega(z_{0})\), \(m\geq 1\).
In this paper, we consider a related problem. We establish a relation between \(w(z)\) and \(zw'(z)\) at the point \(z_{0}\) such that \(|w(z_{0})|=\min \{|w(z)|:|z|= |z_{0}| \}\) or at the point \(z_{0}\) satisfying (1.1). We consider the p-valent functions.
Lemma 1.2
Proof
If we additionally assume that \(w(z)/z^{p}\) is univalent in the unit disc, then we have the following result.
Remark 1.3
2 Applications
For \(p=0\), then Lemma 1.2 becomes the following corollary.
Corollary 2.1
A simple contraposition of Lemma 1.2 provides the following two corollaries.
Corollary 2.2
Corollary 2.3
Proof
Theorem 2.4
Proof
For some other geometrical properties of analytic functions, we refer to the papers [2–4].
3 Conclusion
In this paper, we have presented a correspondence between an analytic function \(f(z)\) and \(zf'(z)\) at the point w, \(0<|w|=R< 1\), in the unit disc \(|z|<1\) on the complex plane such that \(|f(w)|=\min \{|f(z)|: f(z)\in\partial f(|z|\leq R) \}\).
Declarations
Acknowledgements
This work was partially supported by the Centre for Innovation and Transfer of Natural Sciences and Engineering Knowledge, University of Rzeszów.
Authors’ contributions
Both authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have 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
References
- Jack, IS: Functions starlike and convex of order α. J. Lond. Math. Soc. 3, 469-474 (1971) MathSciNetView ArticleMATHGoogle Scholar
- Nunokawa, M, Sokół, J: The univalence of α-project starlike functions. Math. Nachr. 288(2-3), 327-333 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Nunokawa, M, Sokół, J: On some geometric properties of multivalent functions. J. Inequal. Appl. 2015, Article ID 300 (2015) MathSciNetView ArticleMATHGoogle Scholar
- Nunokawa, M, Sokół, J, Cho, NE: Some applications of Nunokawa’s lemma. Bull. Malays. Math. Soc. 40(4), 1791-1800 (2017) MathSciNetView ArticleGoogle Scholar