数理解析研究所講究録
第 1824 巻 2013 年 96-99

96

An extension of Nunokawa lemma
Hitoshi Shiraishi
Abstract
be the class of functions $p(z)=a_{0}+a_{r}z^{n}+\cdots$ which are analytic
Letin the open unit disk . For functions $f(z)$ which are analytic in with $f(O)=1,$68 (1992), 152-153) have shown someM. Nunokawa (Proc. Japan Acad., Ser.
theorems. The object of the present paper is to discuss Nunokawa lemma for theclass$\mathcal{H}[a_{0},n]$$\mathbb{U}$$\mathbb{U}$$A$$\mathcal{H}[a_{0},n].$1IntroductionLet$\mathcal{H}[a_{0}, n]$denote the class of functions $p(z)$ of the form$p(z)=a_{0}+ \sum_{k=n}^{\infty}a_{k^{Z^{k}}}$which are analytic in the open unit disk $\mathbb{U}=\{z\in \mathbb{C} : |z|<1\}$ for somepositive integer$a_{0}\in \mathbb{C}$and a$n.$

The basic tool in proving our results is the following lemma due to S. S. Miller andT. Mocanu [1] (also [2]).

$P.$

Let the function $w(z)$ definded byLemma 1.

$w(z)=a_{m}z^{n}+a_{n+1}z^{n+1}+a_{n+2^{Z^{n+2}+}}\cdots (n=1,2,3, \cdots)$

be analytic inat a point$\mathbb{U}$with $w(O)=0.$ If $|w(z)|$ auains its maximum value on the circle $|z|=r$$\triangleleft)\in \mathbb{U}$, then there enists a real number $m\geqq n$ such that$\frac{z_{0}w’(z_{0})}{w(z_{0})}=m.$2Main result
Applying Lemma 1, we derive the following result.

2010 Mathematics Subject Classification: Primary $30C45$
Keywords and phrases : analytic, univalent, Jack’s lemma, Nunokawa lemma

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Theorem 1. Let
pointsuch that$p(z)\in \mathcal{H}[a_{0}, n]$for some real $a_{0}>0$ and suppose that there exists a$z_{0}\in \mathbb{U}$for
a pure imagenary number for some real${\rm Re}(p(z))>0$

and
Then we have$p(z_{0})=\beta i\dot{w}$$|z|<|z_{0}|$$\beta\neq 0.$$\frac{z_{0}p^{l}(z_{0})}{p(z_{0})}=il$where$l \geqq\frac{n}{2}(\frac{a_{0}}{\beta}+\frac{\beta}{a_{0}})\geqq n$if$\beta>0$

and$l \leqq\frac{n}{2}(\frac{a_{0}}{\beta}+\frac{\beta}{a_{0}})\leqq-n$

if$\beta<0.$

Let us put

Proof.

$w(z)= \frac{a_{\theta}-p(z)}{a_{0}+p(z)}=c_{n}z^{n}+c_{n+1^{Z^{n+1}}}+c_{n+2}z^{n+2}+\cdots (z\in \mathbb{U})$.

Then, we have that $w(z)$ is analytic in $|z|<|z_{0}|,$ $w(O)=0,$ $|w(z)|<1$ for $|z|<|z_{0}|$and$|w(z_{0})|=| \frac{a_{0}^{2}-\beta^{2}-2a_{0}\beta i}{a_{0}^{2}+\beta^{2}}|=1.$

From Lemma 1, we obtain$\frac{z_{0}w’(z_{0})}{w(z_{0})}=\frac{-2a_{0}z_{0}p’(z_{0})}{a_{0}^{2}-\{p(z_{0})\}^{2}}=\frac{-2a_{0}z_{0}p’(z_{0})}{a_{0}^{2}+\beta^{2}}=m (m\geqq n)$.

This shows that$z_{0}p’(z_{0})=- \frac{m}{2}(a_{0}+\frac{\beta^{2}}{a_{0}}) (m\geqq n)$

From the fact that$z_{0}p^{l}(z_{0})$can put

is a real number and$\frac{z_{0}p’(z_{0})}{p(z_{0})}=il$where is a real number.
$l$$p(z_{0})$.

is a pure imaginary number, we98
For the case $\beta>0$ , we have$l={\rm Im}( \frac{z_{0}p’(z_{0})}{p(z_{0})})$$={\rm Im}(- ap^{l}(z_{O})\frac{1}{\sqrt{}}i)$$= \frac{m}{2}(a_{0}+\frac{\beta^{2}}{a_{0}})$$\geqq\frac{n}{2}(a_{0}+\frac{\beta^{2}}{a_{0}})\frac{1}{\beta}$$= \frac{n}{2}(\frac{a_{0}}{\beta}+\frac{\beta}{a_{0}})\geqq n$

and for the case $\beta<0$ , we get$l={\rm Im}( \frac{z_{0}p’(z_{0})}{p(z_{0})})$$={\rm Im}(-z_{0}p’(z_{0}) \frac{1}{\beta}i)$$= \frac{m}{2}(a_{0}+\frac{\beta^{2}}{a_{0}})$$\leqq\frac{n}{2}(a_{0}+\frac{\beta^{2}}{a_{0}})\frac{1}{\sqrt{}}$$= \frac{n}{2}(\frac{a_{0}}{\sqrt{}}+\frac{\beta}{a_{0}})\leqq-n.$This completes our proof.

$\square$Putting $a_{0}=1$ in Theorem 1, we have Corollary 1.
Corollary 1.

Let$p(z)\in \mathcal{H}[1, n]$and suppose that there enists a point${\rm Re}(p(z))>0$

for$|z|<|z_{0}|,$and $p(4)\neq 0.$
Then we have${\rm Re}(p(z_{0}))=0$$\frac{z_{0}p’(z_{0})}{p(z_{0})}=il$where is a real and$l$$|l|\geqq n.$$z_{0}\in \mathbb{U}$such that99

References[1] S. S. Miller and P. T. Mocanu, Second-order differential inequalities in the complex
plane, J. Math. Anal. Appl. 65(1978), 289-305.
[2] S. S. Miller and P. T. Mocanu, Differential Subordinations, Theory and Applications.
Monographs and Textbooks in Pure and Applied Mathematics, 225. Marcel Dekker,

Inc., New York, 2000.

[3] M. Nunokawa, On propeties of non-Camth\’eodory functions, Proc. Japan Acad., Ser.
A 68 (1992), 152-153.

[4] M. Nunokawa, On the order of strongly starlikeness of strongly convex functions, Proc.
Japan Acad. Ser. A Math. Sci. 69(1993), 234-237.

Hitoshi Shiraishi
Department of Mathematics
Kinki University
Higashi-Osaka, Osaka 577-8502
Japan-mail: step-625@hotmail.com$E$