数理解析研究所講究録
第 1626 巻 2009 年 73-75

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On Nunokawa’s Lemma
Mamoru Nunokawa
Abstract
I. S. Jack [1] proved the following lemma 1
Let $w(z)$ be regular in the unit disc, $w(O)=0$ . Then if $w(z)$ attains its msximum value on
the circle $|z|=r$ at a point , then we can write$z_{1}$$z_{1}w’(z_{1})=kw(z_{1})$where is real and $1\leqq k$ . Many and many mathematicians applied the above lemma andobtained numerous interesting results. In this paper, we will obtain a lemma which may be$\infty mected$ intmiately to the Jack’slemma.
$k$1

Basic geometrical property

Property 1Let$\varphi(z)$be analytic in $|z|<1,$$\varphi(z)\neq 0$

in $|z|<1$ and suppose that$\dot{m}n|\varphi(z)|=|\varphi(\triangleleft)|$$|z|\leqq r$

and$\max|\varphi(z)|=|\varphi(z_{1})|$$|z|\leqq r$

where $0<r<1$ and $|*|=|z_{1}|=r$ . Then we have$\frac{\eta\varphi’(\infty)}{\varphi(\triangleleft)}={\rm Re}\frac{\triangleleft\varphi’(\eta)}{\varphi(z_{0})}=(\frac{d\arg\varphi(z)}{d\theta})_{\theta=\theta_{0}}<0$

and$\frac{z_{1}\varphi’(z_{1})}{\varphi(z_{1})}={\rm Re}\frac{z_{1}\varphi’(z_{1})}{\varphi(z_{1})}=(\frac{d\arg\varphi(z)}{d\theta})_{\theta=\theta_{1}}>0$

where$z=re^{i\theta},$$0\leqq\theta<2\pi_{f}\triangleleft$}$=re^{w_{0}}$and$z_{1}=re^{i\theta_{1}}$.

A proof of Property 1 is trivial by considering geometrical property.

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2

Nunokawa’s lemma

Lemma 1 Let$|z_{0}|<1$ such that$\varphi(z)$be analytic in $|z|<1,1<\varphi(0)$ and suppose that there exists a point

for $|z|<|z_{0}|$$1<|\varphi(z)|$$1=|\varphi(z_{0})|$and$\varphi(z_{0})\neq-1$$or$$\min|\varphi(z)|=|\varphi(z_{0})|=1\neq-\varphi(z_{0})$国$\leqq$lzo lThen we have$\frac{z_{0}\varphi’(z_{0})}{\varphi(z_{0})}=ffi\frac{z_{0}\varphi^{l}(z_{0})}{\varphi(z_{0})}\leqq-\frac{\varphi(0)-1}{\varphi(0)+1}$

Proof.

.

.

Let us put(1)

for $|z|<|z_{0}|$ .

$\varphi(z)=\frac{1+p(z)}{1-p(z)}$Then it follows that$p(z)= \frac{\varphi(z)-1}{\varphi(z)+1}$and$0<p(0)= \frac{\varphi(0)-1}{\varphi(0)+1}={\rm Re}\frac{\varphi(0)-1}{\varphi(0)+1}<1$.

From the hypothesis of the Lemma 1 and (1), we havefor $|z|<|z_{0}|$$0<{\rm Re} p(z)$and$0=R\epsilon$p(掬).

Putting$\Phi(z)=\frac{p(0)-p(z)}{p(0)+p(z)}$,$\Phi(0)=0$

and applying the same method as the proof of [2], we have$|\Phi(z)|<1$

for $|z|<|z_{0}|$and$|\Phi(z_{0})|=1$

and therefore$\frac{z_{0}\Phi^{l}(z_{0})}{\Phi(z_{0})}=\frac{-2p(0)\eta p^{f}(z_{0})}{p(0)^{2}-p(a)^{2}}=\frac{-2p(0)a\emptyset(z_{0})}{p(0)^{2}+|p(z_{0})|^{2}}\geqq 1$

It shows that$-z_{0}p’(z_{0}) \geqq\frac{1}{2}(\frac{p(0)^{2}+|p(z_{0})|^{2}}{p(0)})$.

.

$z_{0}$,

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Fkom (1) and Lemma 1, we have$\frac{z_{0}\varphi’(z_{0})}{\varphi(z_{0})}$$={\rm Re} \frac{z_{0}\varphi’(z_{0})}{\varphi(z_{0})}$$=$$\frac{2z_{0}p’(z_{0})}{1-p(z_{0})^{2}}$$=$$\frac{2z_{0}p’(z_{0})}{1+|p(z_{0})|^{2}}$$\leqq$$- \frac{1}{p(0)}(\frac{p(0)^{2}+|p(z_{0})|^{2}}{1+|p(z_{0})|^{2}})$$=$$- \frac{p(0)}{p(0)^{2}}(\frac{p(0)^{2}+|p(z_{0})|^{2}}{1+|p(a)|^{2}})$$=$$-p(0)( \frac{p(0)^{2}+|p(\infty)|^{2}}{p(0)^{2}+p(0)^{2}|p(z_{0})|^{2}})$$<$$-p(0)$$=$$-( \frac{\varphi(0)-1}{\varphi(0)+1})$.

It completes the proof.

口Applications of this lemma will be obtained by me and my hiends.

References[1] I. S. Jack, Ilmctions starlike and convex of order$469rightarrow 474$.

$\alpha$, J. London Math. Soc. (2), 3 (1971),[2] M. Nunokawa, On properties of Non-Carath\’eodory functions, Proc. Japan Acad. Vol. 68, Ser.
$A$ ,No.6 (1992), 152-153.

Mamoru Nunokawa
Emeritus Professor of University of Gunma
Hoshikuki-cho 798-8, Chuou-Ward,
Chiba city $P\theta O- 0808$
Japan
e-mail: mamoru-nuno@doctor.nifty.$jp$