数理解析研究所講究録
第 714 巻 1990 年 18-21

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A note on Bazilevi\v{c} functions

D. K. THOMAS (ウェールズ大学)

INTRODUCTION
For $\alpha>0$ , letdenote the class of Bazilevi\v{c} functions definedin the open unit disc $D=\{z : |z|<1\}$ normalized so that $f(O)=0$ ,$f’(O)=1$ and such that for $z\in D$ ,$B_{1}(\alpha)$${\rm Re} f’(z)( \frac{f(z)}{z})^{\alpha-1}>0$.

(1)This class of functions was studied first by Singh [4] and has been considered recently by several authors e.g. [2,3,5]. We note tliat $B_{1}(1)=R$ ,the class of functions whose derivative has positive real part.
For $f\in R$ , Hallenbeck [1] showed that for $z=re^{i\theta}\in D$${\rm Re} \frac{f(z)}{z}\geq-1+\frac{2}{r}\log(1+r)>-1+2\log 2$,,with equality for
function $f_{1}(z)=-z+2\log(1+z)$ and forthe non-sharp estimate ${\rm Re}(f(z)/z)^{\alpha}>1/(1+2\alpha)$ was obtained in [3]. In
this note, we give the sharp estimate for tlie lower bound ofwhen $f\in B_{1}(\alpha)$ and extend the result to obtain sharp estimates for thereal part of some iterated integral operators.
$t1_{1}e$$B_{1}(\alpha)$${\rm Re}(f(z)/z)^{\alpha}$RES ULTS
For $z\in D$ and $71=1,2,$$\ldots$, define$I_{71}(z)= \frac{1}{z}\int_{0}^{z}I_{?l-\perp}(t)clt$,

where $I_{0}(z)=(f(z)/z)^{\alpha}$ .
Typeset by$A_{\mathcal{M}}S- T_{E\prime}X$

19THEOREM. Let $f\in B_{1}(\alpha)$ an $dz=re^{i\theta}\in D$ . Tlien for $n\geq 0$ ,${\rm Re} I_{n}(z)\geq\gamma_{n}(r)>\gamma_{n}(1)$$vvl_{1}$,ei $e$$0< \gamma_{n}(r)=-1+2\alpha\sum_{k=1}^{\infty}\frac{(-1)^{k+1}r^{k-1}}{k^{n}(k-1+\alpha)}<1$

Equali $ty$ occu

$l’ S$

for tfie $fu$ nction$f_{\alpha}$

defin $ed$ by$f_{\alpha}(z)=( \alpha\int_{0}^{z}t^{\alpha-1}(\frac{1-t}{1+t})dt)^{1/\alpha}$$l^{j}Ve$.

.

note th at wlien $n=0$ ,,${\rm Re}( \frac{f(z)}{z})^{\alpha}\geq\frac{\alpha}{r^{\alpha}}\int_{0}^{r}\rho^{\alpha-1}(\frac{1-\rho}{1+\rho})dp=-1+2\alpha\sum_{k=1}^{\infty}\frac{(-1)^{k+1_{7}.k-1}}{(k-1+c\alpha)}$which reduces $to-1+(2/\uparrow)\log(1+r)$ when $\alpha=1$ .
PROOF: From (1) write$f^{l}(z)( \frac{f(z)}{z})^{\alpha-1}=h(z)$where $h\in P,$ $i.e.,$ $h(0)=1$ and ${\rm Re} l\iota(z)>0$ forThus,$z=\uparrow e^{i\theta}\in D$.

${\rm Re}( \frac{f(z)}{z})^{\alpha}=\alpha{\rm Re}(\frac{1}{z^{\alpha}}\int_{0}^{z}t^{\alpha-1}l\iota(t)dt)$

Write$t=pe^{i\theta}$, so that${\rm Re}( \frac{f(z)}{z})^{\alpha}=\frac{\alpha}{r^{\alpha}}\int_{0}^{r}\rho^{\alpha-1}{\rm Re} h(\rho e^{i0})d\rho$$\geq\frac{\alpha}{r^{\alpha}}\int_{0}^{r}p^{\alpha-1}(\frac{1-\rho}{1+\rho})d\rho$since $h\in P$ .
2,,.

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Hence
${\rm Re} I_{0}(z)={\rm Re}( \frac{f(z)}{z})^{\alpha}\geq\frac{\alpha}{r^{\alpha}}\int_{0}^{r}\rho^{\alpha-1}(\frac{1-\rho}{1+\rho})d\rho$.

Next${\rm Re} L_{z+1}(z)={\rm Re} \frac{1}{z}\int_{0}^{z}I_{n}(t)dt$,$= \frac{1}{r}\int_{0}^{r}{\rm Re} I_{n}(\rho e^{i\theta})dp$,$\geq\frac{1}{r}\int_{0}^{r}(-1+2\alpha\sum_{k=1}^{\infty}\frac{(-1)^{k+1}\rho^{k-1}}{k^{n}(k-1+\alpha)})d\rho$$=\gamma_{n+1}(7^{\cdot})$,,

where the inequality follows by induction.
Now set$\phi_{n}(\uparrow\cdot)=\alpha\sum_{k=1}^{\infty}\frac{(-1)^{k+1}r^{k-1}}{k^{n}(k-1+\alpha)}$.

This series is absolutely convergent for $n\geq 0$ and $0<?$ $<1$ . Suitably
and soshows thatrearranging pairs of terms in.
Finally we note that since for $n\geq 1$$\frac{1}{2}<\phi_{n}(\uparrow)<1$$\phi_{n}(r)$$0<\gamma_{l}(\uparrow\cdot)<1$$r \phi_{n}(r)=\int^{r}\phi_{7l-1}(\rho)d\rho$,induction shows tltat
and sodecreases witlt
for fixed and increases to 1 as
for fixed .
$\phi_{7\lambda}’(r)<0$$7t$$\gamma_{n}(?\cdot)$$7larrow\infty$$\uparrow a’s\uparrow\cdotarrow 1$$\uparrow$

REFERENCES
1. D. J. Hallenbeck, Convex Hulls and Extreme points of some families of univalent functions, Trans. Ainer. Matli. Soc. 192 (1974), 285-292.
2. R. R. London and D. K. Thomas, The derivative of Bazilevi\v{c} functions, Proc.
Amer. Matlt. Soc. 104 (1988), 235-238.

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3. M. Obradovic and S. Owa, Certain subclasses of Bazilevi\v{c} functions of type ,Int. J. Math. and Math. Sci. 9 (1986), 347-359.
4. R. Singh, On Bazilevi\v{c} functions, Proc. Ainer. Matlt. Soc. 38 (1973), 261-271.
5. D. K. Thomas, On a subclass of Bazilevi\v{c} functions, Int. J. Matli. and Math.
Sci. 8 (1985), 779-783.
$\alpha$1980 Mathematics subject classifications: Primary $30C45$

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