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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