数理解析研究所講究録 1341 巻 2003 年 120-125

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An extension of the univalence criteriaof Nehari and Ozaki
Horiana Ovesea-TudorandShigeyoshi Owa

Abstract
In this paper, we obtain a sufgcient condition for the univalence of analyticfunctions in the open
unit disk U. This condition involves two arbitrary functions $g(z)$ and $h(z)$ analytic in U. Replacing$g(z)$ and $h(z)$ by some particular functions, wefind the well-knovn conditio $ns$ for univalency
established by Z.Nehari (Bull. Amer. Math. Soc. 55(1949)) and S. Ozaki (Proc. Amer. Math.
Sco. 33(1972) . Likewise we find other teew sufficient condition$)$1Introduction
We denote by $\mathrm{u}_{r}=\{z \in \mathbb{C} : |z|<r\}$ the disk of 2-plane, where $r\in(0,1]$ ,and. Let $A$ be the class of functions $f(z)$ which are analytic in with the normalizations$f(0)=0$ and $f’(0)=1$ . In the present paper, we consider the following conditions for univalencyof functions $f(z)$ belonging to the class $A$.
$\mathrm{u}_{1}=\mathrm{u}$$I=[0,\infty)$

Theorem 1.1. ([1])$\mathrm{u}$Let $f(z)\in A$ . If, for all$z\in \mathrm{U}_{f}f(z)$$| \{f;z\}\mathrm{j}\leqq\frac{2}{(1-|z|^{2})^{2}}$satisfies

,

(1.1)

where
$\{f;z\}=(\frac{f’(z)}{f’(z)})’-\frac{1}{2}$,$( \frac{f’(z)}{f^{l}(z)})^{2}$

,

(1.2)

then the function $f(z)$ is univalent in U.

Theorem 1.2. ([2])

Let $f(z)\in A$ . If, for all, $f(z)$ satisfies$z\in \mathrm{u}$$| \frac{z^{2}f’(z)}{f(z)^{2}}-1|<1$,(1.3)

then the function $f(z)$ is univalent in U.

2000 Mathematics Subject Classification. Primary.
Key Words and Phrases. Univalent function, L\"owner chain, Nehari criterion, Ozaki criterion$30\mathrm{C}45$

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If we take Koebe functon
for the class of starlike functions in thenExample 1.1.

$f(z)= \frac{z}{(1-z)^{2}}$which is the extremal function$\mathrm{u}_{f}$$(z\in \mathrm{U})$$| \frac{z^{2}f’(z)}{f(z)^{2}}-1|=|-z^{2}|<1$.

2PreliminariesOur considerations are based on the theory of Lowner chains. We first recall here the following
basic result of this theory by Pommerenke.

forLet $L(z,t)=a_{1}(t)z+a_{2}(t)z^{2}+\ldots$ , $a_{1}(t)\neq 0$ be analytic in
Theorem 2.1. ([4]).
Foralmosttorespectwith
all $t\in I$ , locally absolutely continuous in and locally unifom
all $t\in I$ suppose that,$\mathrm{u}_{r}$$\mathrm{u}_{r}$$I_{f}$$(\forall z\in \mathrm{U}_{r})$$z \frac{\partial L(z,t)}{\partial z}=p(z, t)\frac{\partial L(z,t)}{\partial t}$, $t\in I$ . If
where $p(z,t)$ is analytic in and satisfies the condition ${\rm Re} p(z,t)>0$ for all
for each
and $\{L(z,t)/a_{1}(t)\}$ forms a nomal family infor
the function $L(z,t)$ has an analytic and univalent extension to the whole disk U.
$z\in \mathrm{U}$$\mathrm{u}$$\mathrm{u},,$$|a_{1}(t)|arrow\infty$$then_{J}$$t\in I_{l}$$tarrow \mathrm{o}\mathrm{o}$3Main resultsMain theorem of our paper is contained inTheorem 3.1. Let $f(z)\in A$ . If, for some $analyt\dot{1}cfi\iota nctions$ $g(z)=1+b_{1}z+\ldots$ and, the following inequalities$h(z)=c_{0}+c_{1}z+\ldots$ in$\mathrm{u}$$| \frac{f’(z)}{g(z)}-1|<1$(3.1),

and$|( \frac{f’(z)}{g(z)}-1)|z|^{4}+z(1-|z|^{2})|z|^{2}(2\frac{f’(z)h(z)}{g(z)}+\frac{g’(z)}{g(z)})$$+z^{2}(1-|z|^{2})^{2}( \frac{f’(z)h(z)^{2}}{g(z)}+\frac{g’(z)h(z)}{g(z)}-h’(z))$

hold true for all

Proof$z\in \mathrm{u}$$\leqq|z|^{2}$(3.2), then the function $f(z)$ is univalent in U.

Let us consider the function $h_{1}(z,t)$ given by$h_{1}(z,t)=1+(e^{t}-e^{-t})zh(e^{-t}z)$.

and from the malyticity of $h(z)$ in it follows that
we have
For all $t\in I$ andSince $h_{1}(0,t)=1$ , there exists adisk , $0<r<1$ in which$h_{1}(z,t)$ is also analytic in U.
$h_{1}(z,t)\neq 0$ for all $t\in I$ . Then the function $L(z, t)$ defined by$z\in \mathrm{u}$$\mathrm{u}$$e^{-t}z.\in \mathrm{u}$$\mathrm{U}_{r}$$L(z,t)=f(e^{-t}z)+ \frac{(e^{t}-e^{-t})zg(e^{-t}z)}{1+(e^{t}-e^{-t})zh(e^{-t}z)}$

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is analytic in$\mathrm{u}_{\mathrm{r}}$for all $t\in I$ and has the following form$L(z,t)=a_{1}(t)z+a_{2}(t)z^{2}+\ldots$,

where $a_{1}(t)=e^{t}$ , $a_{1}(t)\neq 0$ for all $t\in I$ and.
From the analyticity of $L(z, t)$ in , it follows that there exists anumber
aconstant $K=K(r_{1})$ such that$\lim_{tarrow\infty}|a_{1}(t)|=\infty$$\mathrm{u}_{r}$$|L(z, t)/a_{1}(t)|<K$$(\forall z\in \mathrm{U}_{f}1 , t\in I)$$r_{1}$, $0<r_{1}<r$ , and.

In consequence, the family $\{L(z,t)/a_{1}(t)\}$ is normal in Un. From the analyticity of,$r_{2},0<r_{2}<r_{1}$for all fixed numbers $T>0$ and, there exists aconstant $K_{1}>0$ (that dependson and ) such that$\frac{\partial L(z,t)}{\partial t}$$T$$\mathrm{r}_{2}$$| \frac{\partial L(z,t)}{\partial t}|<K_{1}$$(\forall z\in \mathrm{U}_{r_{2}}, t\in[0, T])$.

It follows that the function $L(z,t)$ is locally absolutely continuous in , locally uniform withrespect to. Let us define the functions $p(z, t)$ and $w(z, t)$ by$I$$\mathrm{u}_{r_{2}}$$p(z,t)=z \frac{\partial L(z,t)}{\partial z}/\frac{\partial L(z,t)}{\partial t}$

and$w(z, t)= \frac{p(z,t)-1}{p(z,t)+1}$.

Then the function $p(z,t)$ is analytic in, $0<r_{3}<r_{2}$ , and the function $p(z, t)$ has an analytic
extension with positive real part in , for all $t\in I$ , if the function $w(z,t)$ can be continued
analytically in and $|w(z,t)|<1$ for all
and $t\in I$ .
After simple computation, we obtain that$\mathrm{u}_{t_{S}}$$\mathrm{u}$$z\in \mathrm{u}$$\mathrm{u}$$w(z,t)=( \frac{f’(e^{-t}z)}{g(e^{-t}z)}-1)e^{-2t}+(1-e^{-2t})e^{-t}z(\frac{2f’(e^{-t}z)h(e^{-t}z)}{g(e^{-t}z)}+\frac{g’(e^{-t}z)}{g(e^{-t}z)})$$+(1-e^{-2t})^{2}z^{2}( \frac{f’(e^{-t}z)h(e^{-\ell}z)^{2}}{g(e^{-t}z)}+\frac{g’(e^{-t}z)h(e^{-t}z)}{g(e^{-t}z)}-h’(e^{-t}z))$

Prom (3.1) and (3.2), we deduce that $g(z)\neq 0$ for all
analytic in U. In view of (3.1) and (3.3), we haveand$w(0, t)=0$$z\in \mathrm{u}$$|w(z,0)|=| \frac{f’(z)}{g(z)}$

.

(3.3)

and then the function $w(z,t)$ is-I $|<1$ .

If $t>0$ is afixed number and, $z\neq 0$ , then the function $w(z,t)$ is analytic in$|e^{-t}z|\leq e^{-t}<1$ for all, and it is known that$z\in \mathrm{u}$(3.4)$\overline{\mathrm{u}}$because$z\in\overline{\mathrm{u}}$$|w(z,t)|= \max|w(\zeta,t)|=|w(e^{i\theta},t)|\mathrm{I}C\mathfrak{l}=1$ ’

Let us denote by$u=e^{-t}e^{i\theta}$. Then$|u|=e^{-t}$$\theta=\theta(t)\in R$

and, from (3.3), we get

.

(3.5)

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$|w(e^{i\theta}, t)|=|( \frac{f’(u)}{g(u)}-1)|u|^{2}+(1-|u|^{2})u(\frac{2f’(u)h(u)}{g(u)}+\frac{g’(u)}{g(u)})$$+(1-|u|^{2})^{2_{\frac{u^{2}}{|u|^{2}}}}( \frac{f’(u)h(u)^{2}}{g(u)}+\frac{g’(u)h(u)}{g(u)}-h’(u))$

and, from (3.4) and (3.5), we conclude, the relation (3.2) implies
Since
that $|w(z,t)|<1$ for all
and $t\in I$ . This gives us that $L(z, t)$ is the L\"owner chain andhence the function $L(z,\mathrm{O})=f(z)$ is univalent in U.
$|w(e^{\dot{*}\theta}, t)|\leq 1$$u\in \mathrm{u}$$z\in \mathrm{u}$$\square$We can get some corollaries for special cases of functions $g(z)$ andas adirect consequence of Theorem 3.1, we get
case$h(z)$. So in the particular$g(z)=\mathrm{f}(\mathrm{z})$Theorem 3.2. Let $f\in A$ .

$If_{f}$$/or$an analytic function $h(z)=c_{0}+c_{1}z+\ldots|.n\mathrm{U}$, $f(z)$ satisfies$|(1-|z|^{2})|z|^{2}(2h(z)+ \frac{f’(z)}{f’(z)})$$+z(1-|z|^{2})^{2}(h(z)^{2}+ \frac{f’(z)h(z)}{f(z)},-h’(z))$

for all$z\in \mathrm{u}$, then the function $f(z)$ is univale

$nt$

(3.6)

$\leqq|z|$

in U.

If we take(3.7)$h(z)=- \frac{1}{2}\frac{f’(z)}{f’(z)}$in Theorem 3.2, then we have
Corollary 3.1. ([1])
If $f(z)\in A$ satisfies the inequality (1.1) for all z
function $f(z)$ is univalent in U.

Proof.

, then the$\in \mathrm{U}$For the function $h(z)$ defined by (3.7), the Schwartzian derivative (1.2) shows that$h(z)^{2}+ \frac{f’(z)h(z)}{f’(z)}-h’(z)=\frac{1}{2}[’\frac{f’(z)}{f’(z)}-\frac{3}{2}(\frac{f’(z)}{f’(z)})^{2}]=\frac{1}{2}\{f;z\}$and then the inequality (3.6) becomes (1.1).
In the particular case $g(z)=( \frac{f(z)}{z})^{2}$ in Theorem 3.1, we have.
$\square$124
Let $f(z)\in A$ . If, for an analytic function $h(z)=c_{0}+c_{1}z+\ldots$ inTheorem 3.3,

s atisfies$\mathrm{u}_{f}\mathrm{f}(\mathrm{z})$(3.8)$| \frac{z^{2}f’(z)}{f(z)^{2}}-1|<1$

and$|( \frac{z^{2}f’(z)}{f(z)^{2}}-1)|z|^{4}+2z(1-|z|^{2})|z|^{2}(\frac{z^{2}f’(z)h(z)}{f(z)^{2}}+\frac{f’(z)}{f(z)}-\frac{1}{z})$$+z^{2}(1-|z|^{2})^{2}[ \frac{z^{2}f’(z)h(z)^{2}}{f(z)^{2}}+2h(z)(\frac{f’(z)}{f(z)}-\frac{1}{z})-h’(z)]|\leqq|z|^{2}$for all z$\in \mathrm{u}_{f}$(3.9)

then the function $f(z)$ is univalent in U.

get Ozaki’s
We remark that the inequality (3.8) is just the inequality (1.3) and we$h(z)$inSo,
wetakeTheorem.
if3.3
function
univalent criterion for aparticular choise of the$\mathrm{w}\mathrm{i}\mathrm{u}$$\mathrm{h}(\mathrm{z})=\frac{1}{z}-\frac{f(z)}{z^{2}}$,(3.10)

then we obtain
Corollary 3.2, ([2])

If $f(z)\in A$ satisfies the $ine\Psi^{lality}(1.3)$ for all z$\in \mathrm{U}$, then thefunction $f(z)$ is univalent in U.
$P$

roof,

For the function $h(z)$ defined by (3.10), we see that$\frac{z^{2}f’(z)h(z)}{f(z)^{2}}+\frac{f’(z)}{f(z)}-\frac{1}{z}=\frac{zf’(z)}{f(z)^{2}}-\frac{1}{z}$and$\frac{z^{2}f’(z)h(z)^{2}}{f(z)^{2}}+2h(z)(\frac{f’(z)}{f(z)}-\frac{1}{z})-h’(z)=\frac{f’(z)}{f(z)^{2}}-\frac{1}{z^{2}}$.

The inequality (3.9) becomes$|( \frac{z^{2}f’(z)}{f(z)^{2}}-1)(|z|^{4}+2|z|^{2}(1-|z|^{2})+(1-|z|^{2})^{2})|\leqq|z|^{2}$,and then$| \frac{z^{2}f’(z)}{f(z)^{2}}-1|\leqq|z|^{2}$.

(3.11)

It is easy to prove that if the inequality (1.3) is true, then the inequality (3.11) is also true.
Indeed, if we put

125$w(z)= \frac{z^{2}f’(z)}{f(z)^{2}}-1$

then the function $w(z)$ is analytic in$\mathrm{u}$,and, since $f(z)\in A$ , we observe that$w(z)=d_{2}z^{2}+d_{3}z^{3}+\ldots$,

which shows that $w(0)=w’(0)=0$ . By inequality (1.3), we have $|w(z)|<1$ . Thus
Schwartz’s lemma gives us that $|w(z)|<|z|^{2}$ .

$\mathrm{t}\mathrm{h}\subset$

Finally, we give aexample for Corollary 3.2.

Example 3.1.

Let us consider the function $f(z)$ given by$f(z)= \frac{z}{1+\sum_{n=1}^{\infty}\frac{1-}{n(n^{2}-1)}z^{n}}$.

Then we have that$\frac{z^{2}f’(z)}{f(z)^{2}}-1=-\sum_{n=1}^{\infty}\frac{1}{n(n+1)}z^{n}$,which gives that$| \frac{z^{2}f’(z)}{f(z)^{2}}-1|<\sum_{n=1}^{\infty}(\frac{1}{n}-\frac{1}{n+1})=1$.

Therefore, the function $f(z)$ is univalent in U.

References[1] Z. Nehari, The Sch wartzian derivate and schlicht functions, Bull. Amer. Math. Soc.
55(1949), 545-551.
[2] S. Ozaki, M. Nunokawa, The Schwartzian derivate and univalent functions , Proc. Amer.
Math. Soc. 33(2), (1972), 392-394.

[3] Ch. Pommerenke,
218(1965), 159-173.

$\tilde{U}\Re r$

die Subordination analytischer Funktionen, J. Reine Angew. Math.,[4] Ch. Pommerenke, Univalent Functions, Vandenhoech Ruprecht in G\"ottingen, 1975.

Horiana Ovesea-Lbdor
Department of Mathematics

Shigeyoshi Owa
Department of Mathematics
Kinki University
Higashi-Osaka, Osaka 577-8502

Romania

Japa“Transilvania” University of Bragov
2200, Bmgov