Conditions for one direction convexity and starlikeness
- Mamoru Nunokawa^{1},
- Oh Sang Kwon^{2},
- Young Jae Sim^{2},
- Ji Hyang Park^{3} and
- Nak Eun Cho^{3}Email author
https://doi.org/10.1186/s13660-016-1034-z
© Nunokawa et al. 2016
Received: 26 January 2016
Accepted: 2 March 2016
Published: 5 March 2016
Abstract
We investigate several sufficient conditions on a function to be convex in one direction or starlike in one direction.
Keywords
MSC
1 Introduction
Let \(\mathcal{H}\) denote the class of functions analytic in the unit disk \(\mathbb{D}:= \{ z \in\mathbb{C}: |z|<1 \}\), and denote by \(\mathcal{A}\) the class of analytic functions in \(\mathcal{H}\) that are normalized by \(f(0)=0=f'(0)-1\). Also, let \(\mathcal{S}\) denote the subclass of \(\mathcal{A}\) composed of functions that are univalent in \(\mathbb{D}\).
We say that a function f is starlike in one direction if f it maps \(|z|=r\) for every r near 1 onto a contour C that is cut by a straight-line passing through the origin in two and no more than two points. Robertson [1] found the following sufficient condition for starlikeness in one direction.
Lemma 1
A function is said to be convex in one direction in \(|z|< r \) (\(r>0\)) if the function maps \(|z|=\rho< r\) for every ρ near r into a contour that may be cut by every straight-line parallel to this direction in no more than two points. It is known (see [1]) that if \(f \in\mathcal{A}\) and \(zf'(z)\) is starlike in one direction, then \(f(z)\) is convex in one direction and belongs to \(\mathcal{S}\). Therefore, we can obtain the following lemma (see also [2–4]).
Lemma 2
We may refer to [5–7] for more sufficient conditions on analytic functions to be convex in one direction.
In the present paper, we investigate several sufficient conditions on functions in \(\mathcal{A}\) to be convex in one direction using various methods. Also, we find sufficient conditions for starlikeness in one direction.
2 Main results
Theorem 1
Proof
Example 1
Theorem 2
Proof
Theorem 3
Proof
Corollary 1
Theorem 4
Proof
Example 2
Applying the same method as that used in the proof of the aforementioned theorems and Lemma 1, we have the following sufficient conditions on analytic functions to be starlike in one direction.
Theorem 5
Theorem 6
Theorem 7
Theorem 8
Declarations
Acknowledgements
The authors would like to express their thanks to the editor Professor S Stevic and the referees for many valuable advices regarding a previous version of this paper. This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2011-0007037).
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
- Robertson, MS: Analytic functions star-like in one direction. Am. J. Math. 58, 465-472 (1936) MathSciNetView ArticleMATHGoogle Scholar
- Ozaki, S: On the theory of multivalent functions, II. Sci. Rep. Tokyo Kyoiku Daigaku, Sect. A 4, 45-86 (1941) MathSciNetMATHGoogle Scholar
- Umezawa, T: On the theory of univalent functions. Tohoku Math. J. 7, 212-228 (1955) MathSciNetView ArticleMATHGoogle Scholar
- Umezawa, T: Analytic functions convex in one direction. J. Math. Soc. Jpn. 4(2), 194-202 (1952) MathSciNetView ArticleMATHGoogle Scholar
- Nunokawa, M, Sokół, J: Some simple conditions for univalence. C. R. Math. Acad. Sci. Soc. R. Can. 354, 7-11 (2015). doi:10.1016/j.crma.2015.10.001 MathSciNetView ArticleGoogle Scholar
- Ogawa, S: Some criteria for univalence. J. Nara Gakugei Univ. 1(10), 7-12 (1961) MathSciNetMATHGoogle Scholar
- Shah, GM: On holomorphic functions convex in one direction. J. Indian Math. Soc. 37, 257-276 (1973) MathSciNetMATHGoogle Scholar