# On a boundary property of analytic functions

## 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

$$\min \bigl\{ \big|w(z)\big|: w(z)\in\partial w\bigl(|z|\leq R\bigr) \bigr\} = \big|w(z_{0})\big|>0.$$
(1.1)

If $$w(z)/z^{p}\neq0$$ in $$|z| < R$$, then

$$\frac{z_{0}w'(z_{0})}{w(z_{0})}=k_{1}\leq p.$$
(1.2)

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

$$\frac{z_{0}w'(z_{0})}{w(z_{0})}=k_{2}\geq p,$$
(1.3)

where $$k_{1}$$, $$k_{2}$$ are real.

### Proof

If

$$\min \bigl\{ \big|w(z)\big|: w(z)\in\partial w\bigl(|z|\leq R\bigr) \bigr\} = \big|w(z_{0})\big|>0,$$

then

$$\bigl\vert w(z) \bigr\vert \geq \bigl\vert w(z_{0}) \bigr\vert \quad\text{for } w(z)\in\partial w\bigl(|z|\leq R\bigr).$$
(1.4)

Then, we also have

$$\biggl\vert \frac{w(z)}{z^{p}} \biggr\vert \geq \biggl\vert \frac{w(z_{0})}{z^{p}_{0}} \biggr\vert \quad\text{for } w(z)\in\partial w\bigl(|z|\leq R\bigr).$$
(1.5)

Let

$$\Phi(z) = w(z)/z^{p}, \quad|z|< 1.$$
(1.6)

Then, from (1.5) and from hypothesis (1.1) we have

$$\min \bigl\{ \big|\Phi(z)\big|:\Phi(z)\in\partial\Phi\bigl(|z|\leq R\bigr) \bigr\} = \big| \Phi(z_{0})\big|.$$
(1.7)

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

$$\min \bigl\{ \big|\Phi(z)\big|: \Phi(z)\in\partial\Phi\bigl(|z|\leq R\bigr) \bigr\} = \big|\Phi(z_{0})\big|,$$
(1.8)

then the function

$$F(z)=\frac{z}{\Phi(z)}=\frac{z^{p+1}}{w(z)}, \quad|z| \leq R,$$

satisfies the assumptions of Jack’s lemma (Lemma 1.1),

$$F(z_{0})=\max_{\theta\in[0,2\pi)} \bigl\{ \big|F(z)\big|: z=Re^{i\theta} \bigr\} ,$$

and hence

$$\frac{z_{0}F'(z_{0})}{F(z_{0})}=p+1-\frac{z_{0}w'(z_{0})}{w(z_{0})}\geq m\geq1.$$

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

$$\frac{\mathrm{d}|\Phi(z)|}{\mathrm{d}\varphi} \bigg\vert _{z=z_{0}}=0.$$
(1.9)

Furthermore, $$\arg \{\Phi(z) \}$$ is increasing at $$z_{0}$$, and so

$$\frac{\mathrm{d}\arg \{\Phi(z) \}}{\mathrm{d}\varphi} \bigg\vert _{z=z_{0}}\geq0.$$
(1.10)

Then we have

\begin{aligned}[b] \frac{z_{0}\Phi'(z_{0})}{\Phi(z_{0})} &= \frac{\mathrm{d}\log\Phi (z)}{\mathrm{d}\log z} \bigg\vert _{z=z_{0}} \\ &= \frac{\mathrm{d}\log|\Phi(z)|+i\mathrm{d}\arg \{\Phi(z) \} }{i\mathrm{d}\varphi} \bigg\vert _{z=z_{0}} \\ &= \frac{\mathrm{d}\arg \{\Phi(z) \}}{\mathrm{d}\varphi}- \frac{i}{|\Phi(z)|}\frac{\mathrm{d}|\Phi(z)|}{\mathrm{d}\varphi} \bigg\vert _{z=z_{0}} \\ &= \frac{\mathrm{d}\arg \{\Phi(z) \}}{\mathrm{d}\varphi} \bigg\vert _{z=z_{0}} \\ &\geq0,\end{aligned}
(1.11)

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

$$\frac{z_{0}w'(z_{0})}{w(z_{0})}=\frac{z_{0}\Phi'(z_{0})}{\Phi(z_{0})}+p.$$
(1.12)

Relations (1.11) and (1.12) imply that

$$\frac{z_{0}w'(z_{0})}{w(z_{0})}\geq p.$$

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

$$\min_{\theta\in[0,2\pi)} \bigl\{ \big|w(z)\big|: z=Re^{i\theta} \bigr\} = \big|w(z_{0})\big|>0.$$
(1.13)

If $$w(z)/z^{p}$$ is univalent and $$w(z)/z^{p}\neq0$$ in $$|z| \leq R$$, then

$$\frac{z_{0}w'(z_{0})}{w(z_{0})}=k_{1}\leq p,$$
(1.14)

where $$k_{1}$$ is real. If $$w(z)/z^{p}$$ is univalent and $$w(z)/z^{p}$$ vanishes in $$|z| \leq R$$, then

$$\frac{z_{0}w'(z_{0})}{w(z_{0})}=k_{2}\geq p,$$
(1.15)

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

$$\min \bigl\{ \big|w(z)\big|: w(z)\in\partial w\bigl(|z|\leq R\bigr) \bigr\} = \big|w(z_{0})\big|>0.$$
(2.1)

If $$w(z)\neq0$$ in $$|z| < R$$, then

$$\frac{z_{0}w'(z_{0})}{w(z_{0})}=k_{1}\leq0.$$
(2.2)

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

$$\frac{z_{0}w'(z_{0})}{w(z_{0})}=k_{2}\geq0.$$
(2.3)

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

$$\min \bigl\{ \big|w(z)\big|: w(z)\in\partial w\bigl(|z|\leq R\bigr) \bigr\} = \big|w(z_{0})\big|>0.$$
(2.4)

If

$$\frac{z_{0}w'(z_{0})}{w(z_{0})}=k_{1}< p$$
(2.5)

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

$$\frac{z_{0}w'(z_{0})}{w(z_{0})}=k_{2}> p,$$
(2.6)

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 $$c\in[0,1)$$,

$$\big|zq'(z)\big|< \frac{p}{c}\big|q(z)\big|^{2},\quad |z| < 1,$$
(2.7)

then the image domain $$q(|z|<1)$$ covers the disc $$|w|< c$$.

### Proof

If

$$\min \bigl\{ \big|q(z)\big|: q(z)\in\partial q\bigl(|z|\leq1\bigr) \bigr\} = \big|q(z_{0})\big|< c,$$
(2.8)

then by (1.2) in Lemma 1.2 we have

$$\frac{z_{0}q'(z_{0})}{q(z_{0})}=k\geq p\quad \Rightarrow \quad\big|z_{0}q'(z_{0})\big| \geq p\big|q(z_{0})\big|.$$
(2.9)

Therefore, by (2.8) and (2.9) we have

$$\big|z_{0}q'(z_{0})\big|\geq\frac{p}{c}\big|q(z_{0})\big|^{2},$$

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

$$\big|p(z) + zp'(z)\big|>c,\quad |z| < 1,$$
(2.10)

where $$c>0$$, and that

$$\mathfrak{Re} \biggl\{ \frac{zp'(z)}{p(z)} \biggr\} >-2, \quad |z| < 1.$$
(2.11)

Then we have

$$\big|p(z)\big|> c, \quad |z| < 1.$$
(2.12)

### Proof

If there exists a point $$z_{0}$$, $$|z_{0}| < 1$$, such that

$$\big|p(z)\big| > c \quad\text{for } |z| < |z_{0}|$$
(2.13)

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

\begin{aligned}[b]\frac{z_{0}p'(z_{0})}{p(z_{0})}&= \frac{\mathrm{d}\log p(z)}{\mathrm{d}\log z} \bigg\vert _{z=z_{0}} \\ &= \frac{\mathrm{d}\log|p(z)|+i\mathrm{d}\arg \{p(z) \} }{i\mathrm{d}\varphi} \bigg\vert _{z=z_{0}} \\ &= \frac{\mathrm{d}\arg \{p(z) \}}{\mathrm{d}\varphi}- \frac{i}{|p(z)|}\frac{\mathrm{d}|p(z)|}{\mathrm{d}\varphi} \bigg\vert _{z=z_{0}} \\ &= \frac{\mathrm{d}\arg \{p(z) \}}{\mathrm{d}\varphi} \bigg\vert _{z=z_{0}} \\ &\leq0.\end{aligned}
(2.14)

From (2.11) and (2.14) we have

$$-2< \frac{z_{0}p'(z_{0})}{p(z_{0})}\leq0,$$

and hence

$$0\leq \biggl\vert 1 + \frac{z_{0}p'(z_{0})}{p(z_{0})} \biggr\vert \leq1.$$

Then it follows that

$$\big|p(z_{0}) + z_{0}p'(z_{0})\big| =\big|p(z_{0})\big| \biggl\vert 1 + \frac{z_{0}p'(z_{0})}{p(z_{0})} \biggr\vert \leq\big| p(z_{0})\big| = c,$$
(2.15)

which contradicts hypothesis (2.10) and therefore completes the proof. □

For some other geometrical properties of analytic functions, we refer to the papers [24].

## 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

1. Jack, IS: Functions starlike and convex of order α. J. Lond. Math. Soc. 3, 469-474 (1971)

2. Nunokawa, M, Sokół, J: The univalence of α-project starlike functions. Math. Nachr. 288(2-3), 327-333 (2015)

3. Nunokawa, M, Sokół, J: On some geometric properties of multivalent functions. J. Inequal. Appl. 2015, Article ID 300 (2015)

4. 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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Both authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript.

### Corresponding author

Correspondence to Janusz Sokół.

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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