On a sufficient condition for starlikeness (Study on Non-Analytic and Univalent Functions and Applications)

On a

sufficient condition

for starlikeness

Mamoru Nunokawa

and Tadayuki

Sekine

Abstract

We give a sufflcient condition for a normalized analytic function to be

starlike.

Keywoids:$\alpha$

-convex

functions, starlikle functions,

convex

functions, univalent

functions

2000

Mathematics Subject

_{Classification.}

Primaly $30C45$

.

1. Introduction and Result

P. T. Mocanu[1] defined$\alpha$

-convex

functions

as

follows: Let $f(z)=z+ \sum_{n=2}^{\infty}a_{n}z^{n}$ be analytic in the unit disc $\Delta=\{z : |z|<1\}$, with $f(z)f’(z)/z\neq 0$ there, and let $\alpha$ be arealnumber. Then $f(z)$ is said to be $\alpha$-convex in $\triangle$ if and onlyif the inequality

${\rm Re}[(1- \alpha)\frac{zf^{f}(z)}{f(z)}+\alpha(1+\frac{zf^{\prime l}(z)}{f^{l}(z)})]>0$

holds in $\triangle$

.

For the functions above,

S. S. Miller, P. T. Mocanu and M. O. Read[2]

proved the following theorem.

Theorm A.

_If

$f(z)=z+ \sum_{n=2}^{\infty}a_{n}z$“ is $\alpha$

-convex

in the unit disc $\Delta$, then$f(z)$

is starlike and univalent in $\triangle$

.

Moreover,

if

$\alpha\geqq 1$, then $f(z)$ _is

convex

for

_$l^{z}l<1$,

and

_if

$\alpha\leqq-1_{f}$ then $1/f(1/z)$ is

convex

for

_$|z|>1$

.

In this paper,

we

partly improve Theorem A and to do so, we need the following lemma[3], [4].

LemmaA. Let$p(z)=1+ \sum_{n=1}^{\infty}c_{\eta}z^{n}$ be analyticin the unit disc$\Delta$ andsupposed

that there enists a point $z_{0}\in\triangle$ such that

${\rm Re}\{p(z)\}>0$

for

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

,

${\rm Re}\{p(z_{0})\}=0$ and $p(z_{0})\neq 0$

.

Then

we

have

$\frac{z_{0}p’(z_{0})}{p(z_{0})}=ik$ 数理解析研究所講究録

when

$k \geqq\frac{1}{2}(a+\frac{1}{a})$ when $\arg\{p(z_{0})\}=\frac{\pi}{2}$

and

$k \leqq-\frac{1}{2}(a+\frac{1}{a})$ when $\arg\{p(z_{0})\}=-\frac{\pi}{2}$

where$p(z_{0})=\pm ia$ and $a>0$.

$Th\infty rem$

.

Let $f(z)=z+ \sum_{n=2}^{\infty}a_{n}z^{n}$ be analytic in the unit disc $\triangle$ and let

us

put

$F(z)=[(1- \alpha)\frac{zf^{f}(z)}{f(z)}+\alpha(1+\frac{zf^{\prime l}(z)}{f^{f}(z)})]$

.

${\rm Re}\{F(z)\}>0$ in $\Delta$,

then $f(z)$ is starlik$e$ in $\triangle$

.

Proof.

For the

case

$\alpha>0$,from the hypothesisofthe theoremwe obtain _{$f’(z)\neq 0$}

in $\Delta$, because ifthere exists a point $z_{0}\in\triangle$ such that

$f’(z_{0})=0$,

this contradicts the hypothesis ofthe theorem.

Let

us

put

$p(z)= \frac{zf^{f}(z)}{f(z)}$ and $p(z)\neq 0$ in $\Delta$

.

Then it follows that

$F(z)=p(z)+ \alpha\frac{zp’(z)}{p(z)}$.

Ifthere exists

a

point $z_{0}\in\Delta$ such that

${\rm Re}\{p(z)\}>0$

for

$|z|<|z_{0}|$,

${\rm Re}\{p(z_{0})\}=0$ and$p(z_{0})\neq 0$

,

then from Lemma $A$, for the

case

_{$\arg\{p(z_{0})\}=\pi/2,$ $p(z_{0})=ia$} and $a>0$, we have $F(z_{0})=p(z_{0})+ \alpha\frac{z_{0}p^{f}(z_{0})}{p(z_{0})}=ia+i\alpha k$

and so, $F(z_{0})$ is apure imaginary value.

Then it followsthat

${\rm Im}\{F(z_{0})\}=a+\alpha k$

$\geqq\frac{1}{2}\{(2+\alpha)c\iota+\frac{\alpha}{a}\}$

$\geqq\sqrt{(2+\alpha)\alpha}$

.

This contradicts the hypothesis of the theorem.

For the

case

$\arg\{p(z_{0})\}=-\pi/2,$ $p(z_{0})=-ia$ and _$a>0$,

we

_{have also}

$F(z_{0})=-ia+i\alpha k$ and ${\rm Im}\{F(z_{0})\}=-a+\alpha k$ $\leqq-a-\frac{1}{2}\alpha(a+\frac{1}{a})$ $=- \{(2+\alpha)a+\frac{\alpha}{a}\}$ $\leqq-\sqrt{(2+\alpha)(x}$

.

This is also contradicts the hypothesis ofthe theorem and it completes the proofof the

case

$\alpha>0$

.

For the case $\alpha<-2$, applying _the same _{method and Lemma} $A$, we can obtain

the proofof the theorem.

Finally, for the

case

$-2\leqq\alpha<0$, it depends

on

[2].

References

[1] P. T. Mocanu, Une propri\’et\’e de convexit\’e g\’en\’eralis\’ee dans la th\’eorie de la

repr\’esentation conforme, Mathematica(Cluj), 11(34), (1969), 127-133, Mr

_42#

7881.

[2] S. S. Miller, P. T. Mocanu andM. O. Read,All $\alpha$

-convex

functions are univalent

and starlike, Proc. Amer. Math. Soc., 37(2), (1973), 553-554.

[3] M. Nunokawa, On properties of Non-Carath\’eodory functions, Proc. Japan

Acad., 68(6), (1992), 152-153.

[4] M. Nunokawa, On the order of stroglystarlikeness ofstrongly

convex

functions, Proc. Japan Acad., 69(7), (1993), 234-237.

798-8, Hoshikuki-machi, Chuo-ku, Chiba-shi, Chiba $2\theta\theta- 0808_{f}$ Japan

E-mail:[email protected]

Research unit

_of

Mathematics, College

_of

Pharmacy, Nihon University

7-1

Namshinodai

$7chome_{f}Flmabashi- shi_{1}$ Chiba 274-8555, Japan

$E\cdot mail$

:[email protected]