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Some criteria for concave conformal mappings
Journal of Inequalities and Applications volume 2015, Article number: 119 (2015)
Abstract
The main purpose of this paper is to derive some criteria for concave conformal mappings.
1 Introduction
A conformal, meromorphic function f on the punctured unit disk
is said to be a concave mapping if \(f(\mathbb{U}^{*})\) is the complement of a compact, convex set.
Let Σ denote the class of analytic functions of the form
then the necessary and sufficient condition for f to be a concave mapping is
where
Recently, Bhowmik et al. [1], Chuaqui et al. [2], Ibrahim and Sokół [3] derived some interesting properties of concave conformal mappings. In this paper, we aim at proving several criteria for the function \(f\in\Sigma\) to be a concave mapping.
To prove our main results, we need the following two lemmas.
Lemma 1.1
(Jack’s lemma [4])
Let \(h(z)=a_{n}z^{n}+a_{n+1}z^{n+1}+\cdots\) be a non-constant analytic function in \(\mathbb{U}\). If \(\vert h(z)\vert \) attains its maximum value on the circle \(\vert z\vert =r<1\), then
where k is a real number with \(k\geq n\).
Lemma 1.2
(See [5])
Let Ω be a set in the complex plane ℂ and suppose that Φ is a mapping from \(\mathbb{C}^{2}\times\mathbb{U}\) to ℂ which satisfies \(\Phi(ix,y;z)\notin\Omega\) for \(z\in\mathbb{U}\) and for all real x, y such that \(y\leq-\frac{1+x^{2}}{2}\). If the function \(p(z)=1+c_{1}z+c_{2}z^{2}+\cdots\) is analytic in \(\mathbb{U}\) and \(\Phi (p(z),zp'(z);z )\in\Omega\) for all \(z\in\mathbb {U}\), then \(\Re(p (z))>0\).
2 Main results
We first give the following result.
Theorem 2.1
Suppose that \(f\in\Sigma\) with \((zf'(z))'\neq 0\). If f satisfies the condition
then f is concave in \(\mathbb{U}^{*}\).
Proof
Assume that
Then the function ϕ is analytic in \(\mathbb{U}\) with \(\phi(0)=0\). From (2.2), we know that
By differentiating both sides of (2.3) with respect to z logarithmically, we get
From (2.1) and (2.4), we find that
Now, we can claim that \(\vert \phi(z)\vert <1\). If not, there exists a point \(z_{0}\in\mathbb{U}\) such that
By Lemma 1.1, we know that
For \(z=z_{0}\), we find from (2.4) and (2.6) that
But (2.7) contradicts (2.5). Thus, we deduce that \(\vert \phi(z)\vert <1\), which implies that
or equivalently,
From (2.9), we get
which shows that the function f is concave in \(\mathbb{U}^{*}\). □
Theorem 2.2
Suppose that \(f\in\Sigma\) with \(f'(z)\neq 0\). If f satisfies the inequality
then f is concave in \(\mathbb{U}^{*}\).
Proof
Define the function \(\varphi(z)\) by
It is easy to see that
is analytic in \(\mathbb{U}\) with \(\varphi(0)=\varphi'(0)=0\). From (2.11), we obtain
Taking logarithmical derivatives of both sides of (2.12) with respect to z, we get
We now show that \(\vert \varphi(z)\vert <1\). If not, there exists a point \(z_{0}\in\mathbb{U}\) such that
By Jack’s lemma, we know that
For \(z=z_{0}\), we have
But (2.15) is a contradiction to condition (2.10), which implies that \(\vert \varphi (z)\vert <1\). Consequently, we deduce from (2.11) that
which implies that f is concave in \(\mathbb{U}^{*}\). □
Theorem 2.3
Suppose that \(f\in\Sigma\) with \(f'(z)\neq 0\). If f satisfies the condition
then f is concave in \(\mathbb{U}^{*}\).
Proof
Suppose that
Then ψ is analytic in \(\mathbb{U}\). From (2.17), we find that
where
For the real numbers x and y satisfying the condition \(y\leq-\frac{1+x^{2}}{2}\), we know that
Now, we take
then \(\Phi(ix,y;z)\notin \Omega\) for all real x, y such that \(y\leq-\frac{1+x^{2}}{2}\). Furthermore, by virtue of (2.16), we know that \(\Phi (\psi(z),z\psi'(z);z )\in\Omega\). Thus, by Lemma 1.2, we get \(\Re(\psi(z))>0\), which shows that f is concave in \(\mathbb{U}^{*}\). □
Finally, we correct an error of Theorem 2.1 in [3], the condition
in it should be changed into
Theorem 2.4
Suppose that \(f\in\Sigma\) with \(f'(z)\neq 0\). If f satisfies the inequality
then f is concave in \(\mathbb{U}^{*}\).
Proof
Define the function \(\omega(z)\) by
Then ω is analytic in \(\mathbb{U}\) with \(\omega(0)=\omega'(0)=0\). From (2.21), we get
Differentiating both sides of (2.22) logarithmically, we get
Now, we show that \(\vert \omega(z)\vert <1\). If not, there exists a point \(z_{0}\in\mathbb{U}\) such that
By Jack’s lemma, we know that
where \(k\geq2\), but this contradicts (2.20), which implies that \(\vert \omega(z)\vert <1\). Thus, f is concave in \(\mathbb{U}^{*}\). □
References
Bhowmik, B, Ponnusamy, S, Wirths, K-J: Concave functions, Blaschke products, and polygonal mappings. Sib. Math. J. 50, 609-615 (2009)
Chuaqui, M, Duren, P, Osgood, B: Concave conformal mappings and pre-vertices of Schwarz-Christoffel mappings. Proc. Am. Math. Soc. 140, 3495-3505 (2012)
Ibrahim, RW, Sokół, J: A geometric property for a class of meromorphic analytic functions. J. Inequal. Appl. 2014, 120 (2014)
Jack, IS: Functions starlike and convex of order α. J. Lond. Math. Soc. 3, 469-474 (1971)
Miller, SS, Mocanu, PT: Differential subordinations and inequalities in the complex plane. J. Differ. Equ. 67, 199-211 (1987)
Acknowledgements
The present investigation was supported by the National Natural Science Foundation under Grant Nos. 11301008 and 11226088, the Aid Program for Science and Technology Innovative Research Team in Higher Educational Institution of Hunan Province, the Foundation of Educational Committee of Henan Province under Grant No. 15A11006. The authors would like to thank the referees for their valuable comments and suggestions which essentially improved the quality of this paper.
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Wang, ZG., Li, ML. Some criteria for concave conformal mappings. J Inequal Appl 2015, 119 (2015). https://doi.org/10.1186/s13660-015-0640-5
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DOI: https://doi.org/10.1186/s13660-015-0640-5