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On a boundary property of analytic functions

Journal of Inequalities and Applications20172017:298

https://doi.org/10.1186/s13660-017-1575-9

• Accepted: 22 November 2017
• Published:

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.

Keywords

• analytic functions
• meromorphic functions
• univalent functions
• boundary behavior

• 30C45
• 30C80

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

()

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). Figure 1 $$\pmb{\Phi(z)}$$ in the case $$\pmb{\Phi(z)\neq0}$$ .
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) Figure 2 $$\pmb{\Phi(z)}$$ in the case $$\pmb{0\in\Phi(|z|\leq R)}$$ .
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 .

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 $$|f(w)|=\min \{|f(z)|: f(z)\in\partial f(|z|\leq R) \}$$.

Declarations

Acknowledgements

This work was partially supported by the Centre for Innovation and Transfer of Natural Sciences and Engineering Knowledge, University of Rzeszów.

Authors’ contributions

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

Competing interests

The authors declare that they have no competing interests. 