On
rvalently
convex
and starlike functi’ons of
order
$\alpha$
MAMORU
NUNOKAWA
Abstract. The
object
of
the present paper is
to give the order of
p–valently
starlikeness for p-mlently
convex. ftulctions of order
$\alpha$in the open unit disk
$U$
.
1
$\mathrm{I}^{\wedge}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$Let
$\cdot$$A(p)$
be the
class
of
functions
$f(z)$
of the form
$f(z)=z^{p}+ \sum_{n=\mathrm{p}+1}^{\infty}a_{n^{Z^{\gamma}}}$
.
which
are
analytic
in the
open
unit disk
$U=\{z\in \mathbb{C}:|z|<1\}$
.
For
$0\leq-\alpha<p$
,
if
$f(z)\in A(p)$
satisfies the following. condition
${\rm Re} \{\frac{zf’(z)}{f(,z\rangle}.\}>\alpha$
$(z\in^{\backslash }U\rangle$
,
then
$f(\#$
is
$\mathrm{s}\mathrm{a}\mathrm{i}\mathrm{d}\cdot \mathrm{t}\mathrm{o}\cdot \mathrm{b}\mathrm{e}$$p$
-vaJently
starlike of
order
$\alpha’,$ $\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{d},\mathrm{b}\mathrm{y}S_{p}^{\dot{\mathrm{f}}}(\alpha)\mathrm{a}\mathrm{n}\mathrm{d},\mathrm{i}\mathrm{f}f\cdot(z\acute’\in A\{p)$ $\mathrm{s}_{\dot{C}}\iota \mathrm{t}\mathrm{i}\mathrm{s}\mathrm{f}\mathrm{i}\mathrm{C}^{\backslash }\mathrm{s}^{\backslash }\mathrm{t}11\mathrm{t}^{\backslash }\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{I}1$$1’+{\rm Re} \{\frac{zf’’(z)}{f’\not\in z^{\sim})},\}>\alpha$
$(z\in U)$
,
then
$f(z\rangle$
is said to
be
$p$
-valently
convex
of order
$\alpha$,
denoted by
$C_{\mathrm{p}}(\alpha)$
.
Jack
[3]
obtained the following
interesting theorem:
If
$f(z)\in C_{1}(\alpha)$
, then
$f(z)\in S_{1^{-}}^{*}(\beta)$
where
$\beta\geq\frac{2\alpha-1+\frac{9-4\alpha+4\alpha^{2}}{}}{4}$
.
The above
estimate
by
Jack [1] is not
sharp,
and after this paper,
$\overline{\mathrm{h}}\mathrm{f}\overline{\mathrm{a}}\mathrm{c}\mathrm{G}\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{o}\mathrm{r}[4]\mathrm{a}\mathrm{n}\mathrm{d}^{-}$Wilken and Feng
[Il
settled this
problem,
$\mathrm{t}\mathrm{h}\dot{\mathrm{e}}\mathrm{i}\dot{\mathrm{r}}$result
$\dot{\mathrm{i}}\mathrm{S}$the fotlowing:
If
$f(z)\cdot\in_{-}C_{1}(\alpha)$
,
then
$f(z)\in_{-}S_{1}^{*}.(\beta)$
,
where
Math
$\epsilon m\alpha t\dot{\tau}\alpha$Subject
Classificationl991:
$30\mathrm{C}45$
Key Words and Phrases:
Analytic, p–valently starlike, p–valently
convex
数理解析研究所講究録
$\beta=\{_{1/2\log 2(\mathrm{i}\mathrm{f}\alpha\cdot=1/2)}^{(1-2\alpha)/2^{2-2\alpha}(1-2^{2\alpha-1}\rangle}$
.
(if
$\alpha\neq 1/2$
)
Very recently, Fukui,
Saigo and Ikeda
[2]
obtained the
following
resulC:
If
$f(z)\in C_{p}(\alpha)$
, then
$f(z)\in S_{p}^{*}(\beta)$
, where
$0\leq\beta<p$
and
(m)
for thc
case,
$0\leq\beta<p/2,$
$\beta$must satisfics
$\beta+\frac{\beta}{2(\beta-p)}\leq\alpha$
,
(b)
for the
case,
$pf2\leq\beta<p,\beta$
must satisfies
$\beta+\frac{2(\oint \mathit{3}-p)}{\beta}\leq\alpha$
.
2
Main
theorem
Theorem 1.
If
$f(,\sim.)\in C_{p}(\alpha)$
,
then
$f(z)\in S_{p}^{*}(\beta)_{f}$
where
$\beta=\frac{2\acute{p}+2\overline{\mathrm{c}}\mathrm{v}-1-\sqrt{4p^{2}+4\alpha^{2}+1-8p\alpha-4p-4\alpha}}{4}$
.
Proof.
Let
us
put
$\frac{r_{d}f^{t}(\nearrow_{\vee}\}}{f(z)}=(p-\beta)\frac{1+w(\nearrow)}{1-w(z)}.+\beta=\frac{(p-2\beta)w(z)+p}{1-w(_{\sim}\gamma\}},-$
where
$0\leq\beta<p,$
$w(z)$
is
$\mathrm{a}\mathrm{n}\mathrm{a}\ddagger \mathrm{y}\mathrm{t}i\mathrm{c}$in
$U$
and
$w(\mathrm{O})=0$
.
By the
logarithmic
$\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\dot{\mathrm{n}},$ $\mathrm{w}\dot{\mathrm{e}}$have
1
$+. \frac{zf’’(z)}{f’(z)}=(p-\beta)\frac{1+u\int(z)}{1-w(z)}+\beta+\frac{(p_{-}-\beta)zli)’(z)}{(p-2\beta)w(z)+p}+\frac{z\tau\prime\prime’(z)}{1-w(z)}$
.
If
there
exists a point
$’\sim’ 0,$
$[_{\sim 0}’|<1\lrcorner \mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h}\cdot \mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}$$|w\not\in z)|<1$
$\mathrm{f}\mathrm{o}\mathrm{r}|z\{<|z_{0}|$
$\mathrm{a}\mathrm{I}\mathrm{l}\mathrm{d}$
$\}w(,\prime_{d}\sim_{0})\}=1$
,
then from
$[$
3, Lemma
$1]_{-}$
, we have
$z_{0}w’(z_{0})=kw(z_{0})$
,
$k\geq 1$
.
Therefore, it
follows
that
$1+ \check{\mathrm{R}}\mathrm{e}\{\frac{z_{0}f’’\langle z_{0})}{-f’(z_{0})}\}--\overline{\mathrm{R}}\mathrm{e}\{_{-}(p-\dot{\beta})\frac{1+w(z_{\theta})}{1-w(z_{0})}+\beta\}+\overline{\mathrm{R}}\mathrm{e}\{_{-}\frac{(p-2\beta)kw(z_{0})}{(p-2\beta)w(z_{0})+p}\}+\check{\mathrm{R}}\mathrm{e}\{_{-}\frac{kw(z_{0})}{1-w(z_{0})}\}$
$= \beta+\frac{k}{2}-\frac{k^{\wedge}}{2}{\rm Re}\{\frac{p-(p-2\beta)w(z_{0})}{p+(p-2\sqrt)w(z_{0})}\}-\frac{k}{2}+\frac{k}{2}{\rm Re}\{\frac{1+w(z_{0})}{1-w(z_{0})}\}$
$\leqq\beta-\frac{\beta}{2(p-\beta)}\equiv\frac{(2p-1)\beta-2\beta^{2}}{2(p-\beta)}$
.
Putting
$\alpha=\frac{(2p-1)\beta-2/f^{2}}{2(p-\beta}$
,
then we have
$\mathrm{T}\mathrm{h}_{\grave{1}}\mathrm{s}$
completes
{he
proof
of
our
theorem.
$\square$Remark
$l$
.
In
$[1],[5]$
and
[6],
the following result was obtained:
If
$f(z)\in C_{p}(0),$
$2\leq p_{\dot{f}}$
then
$f’(’z)\in S_{p}^{*}(\mathrm{O}f\mathrm{a}\mathrm{n}\mathrm{d}\cdot \mathrm{t}\mathrm{h}\mathrm{i}\mathrm{s}$result
is
$\overline{\mathrm{s}}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{p}$.
This
paper
can
be the
same
situation
as Jack’s paper
[3]
contributed
to
$\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{G}\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{o}\mathrm{r}[4]$and
Wilken
and
Feng’s
theorem
[7].
The
author
expect
someone
will obtain
an
exact
result for this problem.
References
[1] S.
$\mathrm{R}_{1}\mathrm{k},\mathrm{u}\dot{\mathrm{g}}_{f}$,
On
$p$
-valently
$\alpha$-convex
functions
of
order
$\beta$(in’ Japanese),
$\mathrm{s}_{\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{k}\mathrm{a}i\mathrm{s}\mathrm{e}\mathrm{k}\mathrm{l}\mathrm{k}\mathrm{e}\mathrm{n}\mathrm{k}\mathrm{y}\mathrm{u}\mathrm{s}\mathrm{h}\mathrm{o}}^{arrow}$
, Kyoto Univ.,
$\mathrm{K}\overline{\mathrm{o}}\mathrm{k}\mathrm{y}\mathrm{u}\tau \mathrm{o}\mathrm{k}\mathrm{u}\mathrm{l}012(1997\rangle,20- 24$
.
$\mathrm{f}^{2}\}$
S.
Fukui,
M.
Saigo and A.
$\mathrm{I}\overline{\acute{\mathrm{k}}}\mathrm{e}\mathrm{d}\mathrm{a}$
,
On
$\overline{M}\overline{a}tx- Stro\overline{h}\tilde{h}\tilde{a}ck\overline{e}r’s$
$th^{-}eo\mathrm{r}em$
for
$p$
-valen
$t$
$analyt\dot{i}\cdot c$
functions
(in
$\mathrm{J}\mathrm{a}\mathrm{p}\mathrm{a}\dot{\mathrm{n}}\mathrm{e}s\mathrm{e}$),
$\mathrm{S}^{\frac{}{\mathrm{u}}}\mathrm{r}\mathrm{i}\mathrm{k}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{e}\mathrm{k}\dot{\mathrm{i}}\mathrm{k}\mathrm{e}\dot{\mathrm{n}}\mathrm{k}\mathrm{y}\mathrm{t}\mathrm{I}\mathrm{s}\mathrm{h}\mathrm{o}\cdot$, Kyoto
Univ.,
$\mathrm{K}\overline{\mathrm{o}}\mathrm{k}\mathrm{y}\mathrm{u}\mathrm{r}\mathrm{o}\mathrm{k}\mathrm{u}\mathrm{l}\mathrm{l}\mathrm{l}2$(1999),17-25.
[3]
I.
$\mathrm{s}_{\vee}$. Jack,
$Fur\dot{\iota}ct_{l}’ons$
starlike and convex
of
orde;
$\cdot$$\alpha$
,
J.
LoIldou Math.
Soc.
3(1971),469-474’.
[4]
T. H.
$\mathrm{M}\mathrm{a}\mathrm{c}\mathrm{G}\mathrm{r}\mathrm{e}\mathrm{g}\mathrm{o}\mathrm{r},$A subordination.
for
convex
ftcn.ctions of
order a,
J. London
$\mathrm{M}\mathrm{a}\mathrm{t}\mathrm{l}\dot{\mathrm{l}}$
.
Soc.
$9(197^{t}\acute{\mathrm{o}}\rangle_{\mathrm{s}}^{l_{)}’},30^{\mathrm{r}}-\delta 36$.
[5]
M.
Nunokawa,
On
$mul\acute{t}ival\overline{e}ntl\overline{y}$
convex
and
$starl_{l}^{r}k\prime e$
functiom,
$\overline{\mathrm{M}}\overline{\mathrm{a}}\mathrm{t}\acute{\mathrm{h}}$.
$\mathrm{J}\overline{\mathrm{a}}\mathrm{p}\mathrm{o}\mathrm{n}$
