Estimate of Real Part of Analytic Functions (Inequalities in Univalent Function Theory and Its Applications)

Estimate

of Real

Part

of Analytic

_Functions

Mamoru Nunokawa

At

_first,

the author shows the following theorem,

Theorem

1. Let

$\mathrm{p}(\mathrm{z})$ be analytic in the unit disk $\mathrm{E}=\{\mathrm{z} : |\mathrm{z}| \langle 1\}$

p $(0)=1$ _and suppose _that

(1) $-\mathrm{z}\mathrm{p}’(\mathrm{z})/( \alpha - \mathrm{p}(\mathrm{z}) )$ $\prec$ $2\mathrm{z}/(1-\mathrm{z}^{\mathrm{a}})$ in $\mathrm{E}$

where 1\langle a _and $\prec$

means

the symbol of subordination.

Then

we

have

${\rm Re}$ p(z) $<$ ain E.

Proof. Putting

q(z) $=$ (

a

- p(z)$)/($

a

- 1) , q(0) $=$ 1,

then q(z) _is analytic

in

E. If there exists apoint z0 (z0$\in$_E $)$

such that

${\rm Re}$ q(z) $>$

0

for

|z|

\langle

I

$\mathrm{z}_{0}|$

数理解析研究所講究録 1276 巻 2002 年 54-56

${\rm Re}$ q $(\mathrm{z}_{0})$ $=$ 0

Then from Nunokaw$\mathrm{a}’\mathrm{s}$ result [1] ,

we

have

$\mathrm{z}_{0}\mathrm{q}$ ’ $(\mathrm{z}_{0})/\mathrm{q}(\mathrm{z}_{0})$ $=\mathrm{z}_{0}\mathrm{p}$ ’ $(\mathrm{z}_{0})/(\mathrm{p}(\mathrm{z}_{0}) - \alpha )$ $=$ ik

where $\mathrm{k}$ is real and

$\mathrm{k}\geqq$ (a $+1/\mathrm{a}$ )/2 when $\arg \mathrm{q}(\mathrm{z}_{0})=\pi/2$

and

$\mathrm{k}\leqq$ - (a $+1/\mathrm{a}$ )/2 when $\arg \mathrm{q}(\mathrm{z}_{0})=\pi$ $/2$

$\mathrm{q}(\mathrm{z}_{0})$ $=$ $\pm \mathrm{i}\mathrm{a}$ and 0 $<\mathrm{a}$

.

This contradicts the hypothesis of the theorem and

so

it

completes the proof.

Applying the

same

method

as

the proof of Theorem 1,

we

$\mathrm{c}\mathrm{a}|$

obtain the following theorem

Theorem

2.

Let $\mathrm{p}(\mathrm{z})$ be analytic in $\mathrm{E}$ and suppose that

(2) $\mathrm{z}\mathrm{P}$

’

$(\mathrm{z})/(\mathrm{p}(\mathrm{z}) -\beta )\dashv$ $2\mathrm{z}/(1 - \mathrm{z}^{\mathrm{g}})$ in E.

where $\beta$ $<$

1

and $\prec$ denote the symbol of subordination.

Then

we

have

$\beta$ $<$ ${\rm Re}$ p(z) in E.

Reference

[1] _M. Nunokawa, On properties of Non-Carath\’eodory functions,

Proceedings of the Japan Academy, Vol. 68, Ser. A, No. 6,

152-153

(1992).

Department of Mathematics

University of Gunma

Aramaki, Maebashi, 371-8510, JAPAN

$\mathrm{e}$-mail: nunokaw@edu. gunma-u.

ac.

ip