数理解析研究所講究録
1538 巻 2007 年 74-81

74

Note on certain analytic functions
Shigeyoshi Owa, Toshio Hayami and Kazuo Kuroki
Abstract
be the class of all analytic functions $f(z)$ in the open unit disk U. Fbr, a subclassof is introduced. The object of the present paper is
to discuss some properties of functions $f(z)$ belonging to the class

Let

$A$

$f(z)\in A$$A$$\mathcal{B}_{k}(\alpha,\beta,\gamma)$$B_{k}(\alpha,\beta,\gamma)$

1.

Introduction
Let

$A$

be the class of functions $f(z)$ of form

(1.1)

$f(z)=z+. \sum_{l=2}^{\infty}a_{n}z^{n}$which are analytic in the open unit disk
to be a member of the subclass$\mathrm{U}=\{z\in \mathbb{C}:|z|<1\}$$B_{k}(\alpha,\beta,\gamma)$

(1.2)

of

$A$

${\rm Re}\{\alpha f^{\langle k)}(z)+\beta zf^{\{k+1)}(z)\}>\gamma$

if it satisfies. A function $f(z)\in A$ is said$(k\in \mathrm{N}=\{1,2,3, \ldots\};z\in \mathrm{U})$

,a
, and
for some1). We. consider some properties for functions $f(z)$ belonging to the class$a_{\mathrm{j}}\in \mathrm{R}(j=2,3,4, \ldots,k)$Remark 1.
Because, for$B_{k}(\alpha,\beta,\gamma)$

is convex.
and$f(z)\in B_{k}(\alpha,\beta,\gamma)$$\in \mathbb{R},\beta\in \mathrm{R}(\beta\neq 0)$$g(z)\in B_{k}(\alpha,\beta,\gamma)$$F(z)=(1-t)f(z)+tg(z)$$\gamma$$\in \mathbb{R}(0\leqq\gamma<k!\alpha a_{kj}a_{1}=$$B_{k}(\alpha,\beta,\gamma)$, we define$(0\leqq t\leqq 1)$.

Then${\rm Re}\{\alpha F^{(k)}(z)+\beta F^{(k+1)}(z)\}$$={\rm Re}\{\alpha(1-t)f^{\langle k)}(z)+\alpha tg^{\langle k)}(z)+\beta(1-t)zf^{\langle k+1)}(z)+\beta tzg^{\{k+1)}(z)\}$$=(1-t){\rm Re}\{\alpha f^{(k)}(z)+\beta zf^{(k+1)}(z)\}+t{\rm Re}\{\alpha g^{(k)}(z)+\beta zg^{(k+1)}(z)\}$$>(1-t)\gamma+t\gamma=\gamma$

Therefore$F(z)\in B_{k}(\alpha,\beta,\gamma)$.

, that is,$B_{k}(\alpha,\beta,\gamma)$

is convex.

2000 $Mathemat$; Subject Classification: Primary.
Kewwods and Phruses: Analytic function, Carathe’odory function.
$30\mathrm{C}45$.

75
In the present paper, we consider some properties of functions $f(z)$ belonging to the class$\mathcal{B}_{k}(\alpha,\beta,\gamma)$2.

Properties of the class$B_{1}(\alpha, \beta, \gamma)$

and$\mathcal{B}_{2}(\alpha,\beta, \gamma)$

We begin with the statement and the proof of the following result.
For cases $k=1$ , we obtain

Theorem 1.
(2.1)

$A$

fimction$f(z\rangle$$\in A\dot{u}$in the class of$B_{1}(\alpha,\beta,\gamma)$

if and only if$f(z)=z+2( \alpha-\gamma)\int_{|\mathrm{g}|=1}(\sum_{n=2}^{\infty}\frac{1}{n((n-1)\beta+\alpha)}x^{n-1}z^{n})d\mu(x)$where $\mu(x)$ is the probability measure on $X=\{x\in \mathbb{C}:|x|=1\}$ .

Proof

For $f(z)\in A$ , we define

(2.2)

Then

$p(z)= \frac{\alpha f’(z)+\beta zf’’(z)-\gamma}{\alpha-\gamma}$$\mathrm{p}(z)$.

is Carath\’eodory function. Therefore we can write(2.3)$\frac{\alpha f’(z)+\beta zf’’(z)-\gamma}{\alpha-\gamma}=\int_{|x|=1}\frac{1+xz}{1-xz}d\mu(x)$

(see

[1]).

It follows from (2.3) that
(2.4)

$z^{p-1} \propto(\frac{\alpha}{\beta}f’(z)+zf’’(z))$$=$$\frac{1}{\beta}z8^{-1}\{\gamma+(\alpha-\gamma)\int_{|oe\mathrm{I}=1}\frac{1+xz}{1-xz}d\mu(x)\}$$\frac{1}{\beta}z^{\frac{a}{\beta}-1}\{\gamma+(\alpha-\gamma)\int_{1oe|=1}(1+xz)(1+xz+x^{2}z^{2}+\ldots)d\mu(x)\}$Integrating the both sides of (2.4), we know that$\int_{0}^{z}\zeta^{\alpha}f^{-1}(\frac{\alpha}{\beta}f’(\zeta)+\zeta f’’(\zeta))d\zeta=\frac{1}{\beta}\int_{\mathrm{I}oe\mathrm{I}=1}\{\int_{0}^{z}(\alpha\zeta^{\alpha}7^{-1}+2(\alpha-\gamma)\sum_{n=1}^{\infty}x^{n}\zeta^{n+\S-1})d\zeta\}d\mu(x)$that is, that$z^{f}f’(z)= \frac{1}{\beta}\int_{|ae\mathrm{I}=1}\{\beta z5+2(\alpha-\gamma)(\sum_{n=1}^{\infty}\frac{\beta}{n\beta+\alpha}x^{n}z^{n+\S)}\}d\mu(x)$$z^{\alpha}p+2( \alpha-\gamma)z^{\alpha}p\int_{|x|=1}(\sum_{n=1}^{\infty}\frac{1}{n\beta+\alpha}x^{n}z^{n})d\mu(x)$Thus, we have

(2.5)$f’(z)=1+2( \alpha-\gamma)\int_{|\sim \mathrm{I}=1}(\sum_{n=1}^{\infty}\frac{1}{n\beta+\alpha}x^{n}z^{n})d\mu(x)$

.

.

,

.

76
An integration of both sides in (2.5) gives us that,$\int_{0}^{z}f’(\zeta)d\zeta=\int_{0}^{z}\{1+2(\alpha-\gamma)\int_{|x|=1}(\sum_{n=1}^{\infty}\frac{1}{n\beta+\alpha}x^{n}\zeta^{n})d\mu(x)\}d\zeta$or$f(z)=z+2( \alpha-\gamma)\int_{|l|=1}(\sum_{n=1}^{\infty}\frac{1}{(n+1)(n\beta+\alpha)}x^{n}z^{n+1})d\mu(x)$.

$=z+ \mathit{2}(\alpha-\gamma)\int_{|oe|-arrow 1}(\sum_{n=2}^{\infty}\frac{1}{n((n-1)\beta+\alpha)}x^{n-1}z^{n})d\mu(x)$This completes the proof of Theorem 1.
The extoeme points of

CoroUary 1.

$\mathcal{B}_{1}(\alpha,\beta,\gamma)$are$(|x|=1)$$f_{\mathrm{g}}(z)=z+ \mathit{2}(\alpha-\gamma)\sum_{n=2}^{\infty}\frac{x^{n-1}}{n((n-1)\beta+\alpha)}z^{n}$

In view of Theorem 1, we have the

If

CoroUary 2.

$f(z\rangle$$\in A$$\mathrm{f}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g}$is in the classcorollary for$\mathcal{B}_{1}(\alpha,\beta,\gamma)$.

, then$(n=\mathit{2},3,4, \ldots)$$|a_{n}| \leqq\frac{2(\alpha-\gamma)}{n((n-1)\beta+\alpha)}$$B\varphi_{l}ahty$$a_{n}$.

.

holds for the $\hslash nctionf(z)$ given by$f(z)=z+2( \alpha-\gamma)\sum_{\mathfrak{n}=2}^{\infty}\frac{x^{n-1}}{n((n-1)\beta+\alpha)}z^{n}$Rrther, the following distortion inequality

If $f(z)\in A$ is in the class

CoroUary 3.

$\mathrm{f}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{s}$If$\mathcal{B}_{1}(\alpha,\beta,\gamma)_{f}$$\beta>0\mathrm{t}\mathrm{d}\frac{\alpha}{\beta}=j(j=2,3,4, \ldots)$.

from Theorem 1.
then$|f(z \rangle|\leqq|z|+2(\alpha-\gamma)(\sum_{n=2}^{\infty}\frac{|z|^{n}}{n((n-1)\beta+\alpha)})$Remark 2.

$(|x|=1)$$(z\in \mathrm{U})$.

in CorolM 3, then we see that$\sum_{n=2}^{\infty}\frac{|z|^{n}}{n((n-1)\beta+\alpha)}\leqq\frac{|z|^{2}}{\beta}\sum_{n=2}^{\infty}\frac{1}{n(n+j-1\rangle}$$= \frac{|z|^{2}}{\beta(j-1)}\sum_{n_{\overline{\sim}}2}^{\infty}(\frac{1}{n}-\frac{1}{n+j-1})$

77$= \frac{|z|^{2}}{\beta(j-1)}\sum_{n=2}^{j}\frac{1}{n}<\frac{\log(j)}{\beta(j-1)}|z|^{2}$.

Therefore, we have that$|f(z)|<|z|+ \frac{2(\alpha-\gamma)\log(j)}{\beta(j-\mathrm{l})}|z|^{2}$$<1+ \frac{2(\alpha-\gamma)\log(j)}{\beta(j-\mathrm{l})}$.

Next, for cases $k=2$ we show

Theorem 2.

A function $f(z)\in A\dot{u}$ in the class & \alpha ,\beta ,$($$\gamma)$

if and only if$f(z)=z+a_{2}z^{2}+2(2 \alpha a_{2}-\gamma)\int_{|x|=1}(\sum_{n=s}^{\infty}\frac{x^{n-2}}{n(n-1)((n-2)\beta+\alpha)}z^{n})d\mu(x)$where

Proof.

$\mu(x)$is the probability measure on $X=\{x\in \mathbb{C}:|x|=1\}$ .

For $f(z)\in A$ , we define$p(z)= \frac{\alpha f’’(z)+\beta zf’’’(z)-\gamma}{2\alpha a_{2}-\gamma}$.

Then $p(z)$ is Carath\’eodory function. Hence, we can write

.

(2.6)

$\frac{\alpha f’’(z)+\beta zf’’’(z)-\gamma}{2\alpha a_{2}-\gamma}=\int_{|x|=1}\frac{1+xz}{1-xz}d\mu(x)$In view of (2.6), we have that
(2.7)$z^{\alpha}7^{-1}( \frac{\alpha}{\beta}f’’(z)+zf^{m}(z))$$=$$\frac{1}{\beta}z^{a}\pi^{-1}\{\gamma+(2\alpha a_{2}-\gamma)\int_{|ae\mathfrak{l}=1}(1+2\sum_{n=1}^{\infty}x^{n}z^{n})d\mu(x)\}$$\frac{1}{\beta}\int_{|\mathrm{r}|=1}(2\alpha a_{2}z^{\frac{\alpha}{\beta}-1}+2(2\alpha a_{2}-\gamma)\sum_{n=1}^{\infty}X^{n_{Z}n+_{p}^{\alpha}-1)d\mu(x)}$Integrating the both sid es of (2.7), we have that‘$\int_{0}$$\zeta^{f-1}(\frac{\alpha}{\beta}f’’(\zeta)+\zeta f^{n\prime}(\zeta))d\zeta$$= \frac{1}{\beta}\int_{|\iota|=1}\{\int_{0}^{z}(2\alpha a_{2}\zeta^{a}F^{-1}+2(2\alpha a_{2}-\gamma)(\sum_{n=1}^{\infty}x^{n}\zeta^{n+_{F}^{a}-1)})d\zeta\}d\mu(x)$,that is, that$z^{a}Tf’’(z)= \frac{1}{\beta}\int_{|\mathrm{r}|=1}\{2\beta a_{2}z^{\alpha}r+2(\mathit{2}\alpha a_{2}-\gamma)(\sum_{n=1}^{\infty}\frac{\beta}{n\beta+\alpha}x^{n}z^{n+_{l)}^{\alpha}}\mathrm{I}^{d\mu(x)}\cdot$.

78This implies that
(2.8)$f”(z)= \int_{|x|=1}\{2a_{2}+2(2\alpha a_{2}-\gamma)(\sum_{n=1}^{\infty}\frac{x^{n}}{n\beta+\alpha}z^{n})\}d\mu(x)$.

An integration of both sides in (2.8) gives us that$\int_{0}^{z}f’’(\zeta)d\zeta=\int_{0}^{z}\{\mathit{2}a_{2}+2(2\alpha a_{2}-\gamma)\int_{|x|=1}(\sum_{n=1}^{\infty}\frac{x^{n}}{n\beta+\alpha}\zeta^{n})d\mu(x)\}d\zeta$or.

$f’(z)-1=2a_{2}z+2(2 \alpha a_{2}-\gamma)\int_{|\iota|=1}(\sum_{n=1}^{\infty}\frac{x^{n}}{(n+1)(n\beta+\alpha)}z^{n+1})d\mu(x)$Therefore, we know that

(2.9)$f’(z)=1+ \mathit{2}a_{2}z+2(\mathit{2}\alpha a_{2}-\gamma)\int_{|ae1=1}(\sum_{n=2}^{\infty}\frac{x^{n-1}}{n((n-1)\beta+\alpha)}z^{n})d\mu(x)$.

Applying the same method for (2.9), we see that$\int_{0}^{z}f’(\zeta)d\zeta=\int_{0}^{z}\{1+2a_{2}\zeta+2(2\alpha a_{2}-\gamma)\int_{|ae1=1}(\sum_{n=2}^{\infty}\frac{x^{n-1}}{n((n-1)\beta+\alpha)}\zeta^{n})d\mu(x)\}d\zeta$

Thus, we obtain that$f(z)=z+a_{2}z^{2}+2(2 \alpha a_{2}-\gamma)\int_{|\mathrm{r}|=1}(\sum_{n=2}^{\infty}\frac{x^{n-1}}{(n+1)n((n-1)\beta+\alpha)}z^{n+1})d\mu(x)$$=z+a_{2}z^{2}+2( \mathit{2}\alpha a_{2}-\gamma)\int_{|ae\mathrm{I}=1}(\sum_{n=\}^{\infty}\frac{x^{n-2}}{n(n-1)((n-\mathit{2})\beta+\alpha)}z^{n})d\mu(x)$This completes the proof of Theorem 2.
Corollary 4.

The extreme points of$B_{2}(\alpha,\beta,\gamma)$are$f_{l}(z)=z+a_{2}z^{2}+2(2 \alpha a_{2}-\gamma)(\sum_{n=3}^{\infty}\frac{x^{n-2}}{n(n-1)((n-\mathit{2})\beta+\alpha)}z^{\mathfrak{n}})$In view of Theorem 2, we have the following corollary for
CoroUary 5.

If$f(z)\in A\dot{u}$in the class$B_{2}(\alpha,\beta,\gamma)$$|a_{n}| \leqq\frac{\mathit{2}(2\alpha a_{2}-\gamma)}{n(n-1)((n-2)\beta+\alpha)}$$a_{n}$$(|x|=1)$.

$(|x|=1)$.

.

, then$(n=3,4,5, \ldots)$.

Equality holds for the function $f(z)$ given by$f(z)=z+a_{2}z^{2}+2( \mathit{2}\alpha a_{2}-\gamma)(\sum_{n=\theta}^{\infty}\frac{x^{n-2}}{n(n-1\rangle((n-\mathit{2})\beta+\alpha)}z^{n})$.

79Further, the following distortion inequality follows from Theorem 2.

Corollary 6.

If$f(z\rangle$$\in A$is in the class$B_{2}(\alpha,\beta,\gamma)_{J}$then$|f(z)| \leqq|z|+|a_{2}||z|^{2}+2(2\alpha a_{2}-\gamma)(\sum_{n=3}^{\infty}\frac{|z|^{n}}{n(n-1)((n-2)\beta+\alpha)})$Properties of the class3

For cases$k$$(z\in \mathrm{u})$.

$B_{k}(\alpha,\beta,\gamma)$

is any natural number, we have

Theorem 3.

$A$fimction $f(z)\in A$ belongs to the class$B_{k}(\alpha,\beta,\gamma)$

if and only if$f(z)=z+a_{2}z^{2}+ \cdots+a_{k}z^{k}+2(k!\alpha a_{k}-\gamma)\int_{\mathrm{I}ae\mathfrak{l}=1}(\sum_{n=k+1}^{\infty}\frac{x^{n-k}z^{n}}{n(n-1)\ldots(n-k+1)((n-k)\beta+\alpha)})d\mu(x)$for $k=1,2,3,$
Pmof.

$\ldots$, where$\mu(x)$; the prvbability measure on $X=\{x\in \mathbb{C}:|x|=1\}$ .

For $f(z)\in A$ , we define$p(z)= \frac{\alpha f^{\langle k)}(z)+\beta zf^{(k+1)}(z)-\gamma}{k!\alpha a_{k}-\gamma}$.

Since $p(z)$ is Carath\’eodory function, we can write that

.

(3.1)

$\frac{\alpha f^{(k)}(z)+\beta zf^{\{k+1)}(z)-\gamma}{k!\alpha a_{k}-\gamma}=\int_{|ae\mathfrak{l}=1}\frac{1+xz}{1-xz}d\mu(x)$This means that
(3.2)

$z5^{-1}( \frac{\alpha}{\beta}f^{(k)}(z)+zf^{(k+1)}(z))$$=$$\frac{1}{\beta}z8^{-1}\{\gamma+(k!\alpha a_{k}-\gamma)\int_{|\mathrm{g}|=1}(1+2\sum_{n=1}^{\infty}x^{n}z^{n})d\mu(x)\}$$\frac{1}{\beta}\int_{|ae1=1}(k!\alpha a_{k}z^{\alpha}p-1+2(k!\alpha a_{k}-\gamma)\sum_{n=1}^{\infty}X^{n_{Z}n+_{l}^{a}-1)d\mu(x)}$Integrating the both sides of (3.2), we obtain that$\int_{0}^{z}\zeta^{\alpha}\pi^{-1}(\frac{\alpha}{\beta}f^{\langle k)}(\zeta)+\zeta f^{\langle k+1)}(\zeta))d\zeta$$= \frac{1}{\beta}\int_{|ae\models 1}\{\int_{0}^{\iota}(k!\alpha a_{k}\zeta^{\S-1}+2(k!\alpha a_{k}-\gamma)(_{n=1}\sum^{\infty}x^{n}\zeta^{\hslash}+_{7}^{\alpha}-1))d\zeta\}d\mu(x)$

,

.

80
that is, that$z^{\alpha}Ff^{(k)}(z)= \frac{1}{\beta}\int_{|x|=1}\{k!\beta a_{k}z^{a}F+2(k!\alpha a_{k}-\gamma)(\sum_{n=1}^{\infty}\frac{\sqrt}{n\beta+\alpha}x^{n}z^{\alpha}p-1)\}d\mu(x)$.

This is equivalent to(3.3)$f^{(k)}(z)= \int_{|\mathrm{r}|=1}\{k!a_{k}+2(k!\alpha a_{k}-\gamma)(\sum_{n=1}^{\infty}\frac{x^{n}}{n\beta+\alpha}z^{n})\}d\mu(x)$Now, since$f(\mathrm{O})=0,$ $f’(\mathrm{O})=1$.

, and $f^{\{m)}(0)=m!a_{m}(m=2,3,4, \ldots)$ , we see that$\int_{0}^{z}f^{(m)}(\zeta)d\zeta=f^{\langle m-1)}(z)-f^{\langle m-1)}(0)$$=f^{(m-1)}(z)-(m-1)!a_{m-1}$.

Furthermore, we know that$\int_{0}^{z}\int_{0}^{\zeta_{n}}\cdots\int_{0}^{\zeta_{2}}m!a_{m}d\zeta_{1}d\zeta_{2}\ldots d\zeta_{m}=a_{m}z^{m}$

,

and

.

$\sum_{n=1}^{\infty}\frac{x^{n}z^{n+k}}{(n+k)(n+k-1)\ldots(n+1)(n\beta+\alpha)}=\sum_{n=k+1}^{\infty}\frac{x^{n-k_{Z^{\hslash}}}}{n(n-1)\ldots(n-k+1)((n-k)\beta+\alpha)}$Therefore, integrating$k$times the both sides in (3.3), $we$ obtain that$\int_{0}^{\epsilon}r_{0}^{\mathrm{k}}\cdots\int_{0}^{\zeta}’ f^{\{k)}(\zeta_{1})d\zeta_{1}d\zeta_{2}\ldots d\zeta_{k}$$= \int_{0}^{z}\int_{0}^{b}\cdots\int_{0}^{\zeta}’\{k!a_{k}+2(k!\alpha a_{k}-\gamma)\int_{|l|=1}(\sum_{n=1}^{\infty}\frac{x^{n}\zeta_{1}^{n}}{n\beta+\alpha})d\mu(x)\}d\zeta_{1}d\zeta_{2}\ldots d\zeta_{k}$,that is, that$f(z)=f(0)+ \int_{0}^{z}f’(0)d\zeta_{1}+\int_{0}^{z}\int_{0}^{\zeta}’ f’’(0)d\zeta_{1}d\zeta_{2}+\int_{0}^{f}\int_{0}^{\mathrm{Q}}[_{0}’ f^{\prime\prime l}(0)d\zeta_{1}d\zeta_{2}d\zeta_{l}+\ldots$$+ \int_{0}^{z}\int_{0}^{\zeta \mathrm{t}}\cdots\int_{0}^{\zeta_{2}}\{$.

$k!a_{k}+ \mathit{2}(k!\alpha a_{k}-\gamma)\int_{|\mathrm{r}|=1}(\sum_{n=1}^{\infty}\frac{x^{n}\zeta_{1}^{n}}{n\beta+\alpha})d\mu(x)\}d\zeta_{1}d\zeta_{2}\ldots d\zeta_{k}$Thus, we conclude that$f(z)=z+a_{2}z^{2}+a_{S}z^{S}+\cdots+a_{k}z^{k}$$+ \mathit{2}(k!\alpha a_{k}-\gamma)\int_{|l|=1}(\sum_{n=k+1}^{\infty}\frac{x^{n-k_{Z^{\hslash}}}}{n(n-1)\ldots(n-k+1)((n-k)\beta+\alpha)})d\mu(x)$The proof of Theorem 3 is complete.

.
$\square$81
Corollary 7.

The extreme points of$\mathcal{B}_{\mathrm{k}}(\alpha,\beta,\gamma)$are$f_{x}(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots+a_{k}z^{k}$$+2(k! \alpha a_{k}-\gamma)(\sum_{n=k+1}^{\infty}\frac{x^{n-k}z^{n}}{n(n-1)\ldots(n-k+1)((n-k)\beta+\alpha)})$$(|x|=1)$.

In view of Theorem 3, we see thatIf $f(z)$ belongs to the class

Corollary 8.

$B_{k}(\alpha,\beta,\gamma)$$|a_{n}| \leqq\frac{2(k!\alpha a_{k}-\gamma)}{n(n-1)\ldots(n-k+1\rangle((n-k)\beta+\alpha)}$, then$(n=k+1,k+\mathit{2}, k+3, \ldots)$.

Equality holds for the function $f(z)$ given by$f(z)=z+a_{2}z^{2}+a_{3}z^{S}+\cdots+a_{k}z^{k}$$+ \mathit{2}(k!\alpha a_{k}-\gamma)(\sum_{n=k+1}^{\infty}\frac{x^{n-k}z^{n}}{n(n-1)\ldots(n-k+1)((n-k)\beta+\alpha)})$$(|x|=1)$.

Rrther, the following distortion inequality folows from Theorem 3.

Corollary 9.

If $f(z)$ belongs to the class$B_{k}(\alpha,\beta,\gamma)$, then$|f(z)|\leqq|z|+|a_{2}||z|^{2}+|a_{3}||z|^{3}+\cdots+|a_{k}||z|^{k}$$+ \mathit{2}(k!\alpha a_{k}-\gamma)(\sum_{n=k+1}^{\infty}\frac{|z|^{n}}{n(n-1)\ldots(n-k+1)((n-k)\beta+\alpha)})$$(z\in \mathrm{u})$.

References[1] D. J. Hallenbeck and T. H. MacGregor, Linear Problems and Convexity Techniques inGeometric Function Theory, Monographs and Studies in Mathematics 22, Pitman (1984).

Depanmefu of Mathematics
Kinki University
Higashi-Osaka, Osaka 577.8502
Japan-mail: $owa@math.hnda|.ac.\dot{p}$$E$