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On a boundary property of analytic functions
Journal of Inequalities and Applications volume 2017, Article number: 298 (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.
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
Let \(w(z) = z^{p} + \sum_{n=p+1} ^{\infty}a_{n} z^{n}\) be analytic in \(|z| < 1\). Assume that there exists a point \(z_{0} \), \(|z_{0}|=R<1\), such that
If \(w(z)/z^{p}\neq0\) in \(|z| < R\), then
If the function \(w(z)/z^{p}\) has a zero in \(|z| < R\) and \(\partial w(|z|\leq R)\) is a smooth curve at \(w(z_{0})\), then
where \(k_{1}\), \(k_{2}\) are real.
Proof
If
then
Then, we also have
Let
Then, from (1.5) and from hypothesis (1.1) we have
There are two cases: \(\Phi(|z|< R)\) contains the origin (see Figure 2); and \(\Phi(|z|< R)\) does not (see Figure 1).
First, suppose that \(\Phi(z)\) does not become 0 in \(|z| < R\). If there exists a point \(z_{0}=R\exp(i\varphi_{0})\), \(0\leq\varphi_{0}<2\pi\), \(0 < R< 1\), such that
then the function
satisfies the assumptions of Jack’s lemma (Lemma 1.1),
and hence
This gives (1.2).
For the case \(0\in\Phi(|z|< R)\) (see Figure 2), for \(\Phi(z)\) given in (1.6), we have that \(|\Phi(z)|\) has an extremum at \(z_{0}\), and so
Furthermore, \(\arg \{\Phi(z) \}\) is increasing at \(z_{0}\), and so
Then we have
because of (1.9). On the other hand, by (1.6) we have \(w'(z)=z^{p}\Phi'(z)+pz^{p-1}\Phi(z)\), and hence
Relations (1.11) and (1.12) imply that
Therefore, by (1.11) we obtain (1.3). □
If we additionally assume that \(w(z)/z^{p}\) is univalent in the unit disc, then we have the following result.
Remark 1.3
Let \(w(z) = z^{p} + \sum_{n=p+1} ^{\infty}a_{n} z^{n}\) be analytic in \(|z| < 1\). Assume that there exists a point \(z_{0} \), \(|z_{0}|=R<1\), such that
If \(w(z)/z^{p}\) is univalent and \(w(z)/z^{p}\neq0\) in \(|z| \leq R\), then
where \(k_{1}\) is real. If \(w(z)/z^{p}\) is univalent and \(w(z)/z^{p}\) vanishes in \(|z| \leq R\), then
where \(k_{2}\) is real.
2 Applications
For \(p=0\), then Lemma 1.2 becomes the following corollary.
Corollary 2.1
Let \(w(z) = 1 + \sum_{n=1} ^{\infty}a_{n} z^{n}\) be analytic in \(|z| < 1\). Assume that there exists a point \(z_{0} \), \(|z_{0}|=R<1\), such that
If \(w(z)\neq0\) in \(|z| < R\), then
If the function \(w(z)\) has a zero in \(|z| < R\) and \(\partial w(|z|\leq R)\) is a smooth curve at \(w(z_{0})\), then
A simple contraposition of Lemma 1.2 provides the following two corollaries.
Corollary 2.2
Let \(w(z) = z^{p} + \sum_{n=p+1} ^{\infty}a_{n} z^{n}\) be analytic in \(|z| < 1\) and suppose that there exists a point \(z_{0} \), \(|z_{0}|=R<1\), such that
If
and \(\partial w(|z|\leq R)\) is a smooth curve at \(w(z_{0})\), then \(w(z)/z^{p}\) has no zero in \(|z| \leq R\). If
then the function \(w(z)/z^{p}\) has a zero in \(|z| \leq R\).
Corollary 2.3
Let \(q(z)=z^{p}+\sum_{n=p+1}^{\infty}a_{n} z^{n}\) be analytic in \(|z|\leq1\). Assume that \(q(z)/z^{p}\) has a zero in \(|z|<1\). If for given \(c\in[0,1)\),
then the image domain \(q(|z|<1)\) covers the disc \(|w|< c\).
Proof
If
then by (1.2) in Lemma 1.2 we have
Therefore, by (2.8) and (2.9) we have
which contradicts hypothesis (2.7) and therefore completes the proof. □
Theorem 2.4
Let \(p(z)\) be analytic in \(|z|<1\) with \(p(z) \neq0\), \(|p(0)|>c\), in \(|z| < 1\) and suppose that
where \(c>0\), and that
Then we have
Proof
If there exists a point \(z_{0}\), \(|z_{0}| < 1\), such that
and \(| p(z_{0}) | = c\), then \(p(|z|\leq|z_{0}|)\) has the shape as in Figure 1 and \(\mathrm{d}|p(z)|/\mathrm{d}\varphi\), \(z=re^{i\varphi}\), vanishes at the point \(z=z_{0}\). Therefore, we have
From (2.11) and (2.14) we have
and hence
Then it follows that
which contradicts hypothesis (2.10) and therefore completes the 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 \(|f(w)|=\min \{|f(z)|: f(z)\in\partial f(|z|\leq R) \}\).
References
Jack, IS: Functions starlike and convex of order α. J. Lond. Math. Soc. 3, 469-474 (1971)
Nunokawa, M, Sokół, J: The univalence of α-project starlike functions. Math. Nachr. 288(2-3), 327-333 (2015)
Nunokawa, M, Sokół, J: On some geometric properties of multivalent functions. J. Inequal. Appl. 2015, Article ID 300 (2015)
Nunokawa, M, Sokół, J, Cho, NE: Some applications of Nunokawa’s lemma. Bull. Malays. Math. Soc. 40(4), 1791-1800 (2017)
Acknowledgements
This work was partially supported by the Centre for Innovation and Transfer of Natural Sciences and Engineering Knowledge, University of Rzeszów.
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Nunokawa, M., Sokół, J. On a boundary property of analytic functions. J Inequal Appl 2017, 298 (2017). https://doi.org/10.1186/s13660-017-1575-9
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DOI: https://doi.org/10.1186/s13660-017-1575-9