数理解析研究所講究録
917 巻 1995 年 55-59

55

ON$\mathrm{K}\mathrm{E}\mathrm{R}0\mathrm{r}0\mathrm{R}\mathrm{P}\mathrm{I}\mathrm{l}\mathrm{l}\mathrm{C}\Lambda 1\prime \mathrm{L}\mathrm{Y}\mathrm{l}\mathrm{U}1,\mathrm{T}\mathrm{l}\mathrm{V}\Lambda 1\prime \mathrm{D}\mathrm{N}\mathrm{T}$Naoya Koi$\mathrm{k}\mathrm{e}$FUNCTIONS

尚包 群馬大学)(小池Mamoru Nurlokawa(布川 護 群馬大学)

Hi toshi Sai

Abstract. The$\mathrm{D}^{1\lrcorner}\mathrm{r}\mathrm{K}$)$\mathrm{S}\mathrm{e}$$\iota \mathrm{t}\mathrm{J}\mathrm{h}$斉, 群馬高専)(斎藤of this paper is to derive somemeromorphically multi valent functions in$\mathrm{a}\mathrm{r}\iota \mathrm{r}\mathrm{l}\mathrm{u}\mathrm{l}\mathrm{u}\mathrm{s}$)of certain$\mathrm{e}\Gamma \mathrm{t}\mathrm{i}\mathrm{e}\mathrm{s}$$\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{t}$.

1. IntroductionLet(1. 1)$\Sigma_{1)}$be the class of functions of the form
$1/zt’+\mathrm{a}_{()}$$\mathrm{f}(\mathrm{z})=$$/\mathrm{z}1’ 14$

which are analytic in the annul us$\mathrm{p}\epsilon \mathrm{N}=\{1,2,3, \}$$(l. 2)$. For$\mathrm{f}(\mathrm{z})\epsilon\Sigma_{1\supset},${$\gamma_{d}$...

$\mathrm{k}$$+\mathrm{a}\mathrm{k}|_{1)}17,$: lzl]},$<$

we define the operator

$|$

where
$\mathrm{D}\mathrm{n}|_{1}\cdot \mathrm{l}\mathrm{f}(\mathrm{z})$by$\mathrm{D}\mathrm{l}1|_{t}\cdot {}^{\mathrm{t}}\mathrm{f}(_{7}\lrcorner)=(\mathrm{z}|’ 12\mathrm{r}\mathrm{l}\mathrm{f}(\mathrm{Z})/(_{\backslash }\mathrm{r}1\dagger_{0^{-}1)}!)\mathrm{t}_{11}1_{\mathfrak{l}\iota)}\mathrm{Y}-/\mathrm{z}\uparrow’$$=1/z\mathrm{r}$$|(\mathrm{n}\text{ト_{}\mathrm{P})}\mathrm{a}_{0}/z\mathrm{p}\mathrm{t}\vdash(\mathrm{n}+\mathrm{p})(\mathrm{n}+_{\mathrm{D}}+])\mathrm{a}1/(2 !

$+(\mathrm{n}+\mathrm{p})(\mathrm{n}+\mathrm{p}\mathrm{i}\mathrm{l})$where$\mathrm{D}--$.........

$\mathrm{n}$is an integer and$(\mathrm{n}+\mathrm{k}\mathrm{t}2\mathrm{p}^{-}])\mathrm{a}\mathrm{k}|\mathrm{p}$[,..

\mathrm{z}\mathrm{r}2)\mathrm{i}$$z\mathrm{k}/(\mathrm{k}\}_{\mathrm{P})}$!

$+$$\cdot$..

.

$\mathrm{n}>-\mathrm{p}$Recently, Cho and Nunokawa [1] proved that${\rm Re}${’$\mathrm{z}\mathrm{t}^{1\mathfrak{j}}$implies($\mathrm{D}11$

ト$\mathrm{t}^{\backslash }\mathrm{f}(7_{J})$

)

$’$

}

$<-\mathrm{a}$${\rm Re}\{\mathrm{z}\mathrm{r}\supset|\mathrm{l}(\mathrm{D}\mathrm{n}|\mathrm{l}’\iota_{\mathrm{f}(\mathrm{z})’} \}$$(0\leqq \mathrm{a}<\mathrm{p}.\cdot 1\mathrm{z}1< 1)$(lzl $<1$ )$<-\beta$

where$\beta=(\mathrm{p}+2\mathrm{a}(\mathrm{n}t\mathrm{p}))/(]\mathrm{f}2(\mathrm{n}i_{\mathrm{P}^{))}}$.

56

In the present paper, we show another properties of functions
concerning witll the operator.

,$\mathrm{f}(\mathrm{z})\epsilon\Sigma_{\mathrm{r}}$$\mathrm{D}^{\mathrm{t})1\mathrm{p}}1\mathrm{f}(7_{\lrcorner})$

2. Main results
We need the fol lowing lemma due to Jackl,emma.

Let$\mathrm{w}(\mathrm{z})$be non-constant analytic in

attains its maximum val ue at athen we haveIf

$1\mathrm{w}(\mathrm{z})1$

Zo

where

$\mathrm{k}$is real and$\mathrm{k}\geqq 1$

Theorem 1. If${\rm Re}$for some$\mathrm{a}(\mathrm{a}>\mathrm{p})$)$\{ 7_{\lrcorner}:1\mathrm{z}1<1 \}$$\mathrm{U}=$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{z}_{\mathrm{t}\mathrm{I}}$due to Mi ller and Mocanu

on the circle

with$1\mathrm{z}1=_{\Gamma<1}$$\mathrm{w}(())=]$.

,$(\mathrm{z}_{()})=\mathrm{k}\mathrm{w}(\mathrm{Z}_{0})$.

$\mathrm{f}(\mathrm{z})\in\Sigma_{1^{\mathrm{R}}}$(2. 1)’$\mathrm{w}$$\propto$(or,$\mathrm{L}2\rfloor$sati sfies$\{ \mathrm{z}^{\mathrm{p}\dagger[}(\mathrm{D}^{\mathrm{n}\}_{\lceil}}’ \mathrm{f}(\mathrm{z}))’ \}$$>$$(\mathrm{z}\in \mathrm{U})$$-\mathrm{a}$then,(2. 2)${\rm Re}$$\{ \mathrm{z}^{\mathrm{p}\}\uparrow}(\mathrm{D}^{t1\mathrm{I}\mathrm{p}-1}\mathrm{f}(\mathrm{z},))’\}>$$-\beta$$(\mathrm{z}\epsilon \mathrm{U})$

,

where
($\beta=$. Define the function$\mathrm{P}\mathrm{r}\infty \mathrm{f}$(2. 3)$\mathrm{p}+2\mathrm{a}$$\mathrm{w}(\mathrm{z})$( ト\beta ))/$\mathrm{n}$$(1 +2(\mathrm{n}\text{ト_{}\mathrm{P}}))$.

by$\mathrm{z}^{\mathrm{p}^{1}1}(1)^{111_{t}})1\mathrm{f}(\mathrm{z}))’=((\mathfrak{p}-2\beta)_{\mathrm{W}(_{7)}},-\mathrm{p})/(1+\mathrm{W}(\mathrm{Z}))$$\mathrm{w}(\mathrm{z})\neq-1$,

with$\beta=(\mathrm{p}+2\mathrm{a}(\mathrm{n}+\mathrm{D}))/(1+2(\mathrm{n}\_{\mathrm{P}))}$

Then$\mathrm{w}(\mathrm{z})$is analytic in(2. 4)$\mathrm{U}$and$\mathrm{w}(\mathrm{O})=0$.

. Note that.

$\mathrm{z}(\mathrm{D}^{i1}|\mathrm{r}’-_{1}\mathrm{f}(\mathrm{z}))’=(\mathrm{n}+_{\mathcal{D}})\mathrm{D}^{\mathrm{n}}\}\mathrm{p}\mathrm{f}(\mathrm{z})-(\mathrm{n}+2_{\mathrm{P}})\mathrm{D}^{\mathfrak{s}1^{\}\mathrm{p}}}\mathrm{t}\mathrm{f}(\mathrm{z})$It follows from (2. 3) that(2.5),$(\mathrm{n}+\mathrm{p})\mathrm{z}^{\mathrm{f}}$’$\mathrm{D}^{\mathrm{n}\vdash \mathrm{p}}\mathrm{f}(\mathrm{z})-(\mathrm{n}\vdash 2\mathrm{p})\mathrm{z}^{\mathrm{p}}$$\mathrm{D}^{\mathrm{n}\}\mathrm{p}-}\uparrow \mathrm{f}(\mathrm{z})$$=((\mathrm{p}-2\beta)_{\mathrm{W}(_{\mathrm{Z}})-}\mathrm{P})/(1+\mathrm{W}(\mathrm{z}))$.

Taking the differentiations in both sides of (2. 5), we have$[3.\rceil.$).

57(2. 6)$\mathrm{z}t^{\iota|1}(\mathrm{D}^{\mathrm{r}1}|l\supset \mathrm{f}(\mathrm{z}))’=((\mathrm{p}-2\beta)\mathrm{w}(\mathrm{Z})-\mathrm{p})/(1mathrm{W}(\mathrm{Z}))$$+2(\mathrm{p}-\beta)r,\mathrm{W}$Suppose that there exists a point’$(r_{J})/((\mathrm{n}+_{\mathrm{P}})(1+\mathrm{w}(_{\mathrm{Z})})\mathit{2})$such that$\epsilon 1\mathrm{J}$$\mathrm{z}_{()}$$(\mathrm{w}(\mathrm{z}_{0})\neq-1)$$\max \mathrm{l}\mathrm{w}(\mathrm{z})1=1\mathrm{w}(\mathrm{z}_{0})1$I$\mathrm{z}1$

$\Xi$

I.

,

$\mathrm{z}_{0}|$then. by Lemma, we have
Zo

Therefore, letting’$\mathrm{w}$(Zo)$\mathrm{w}(\mathrm{z}_{0})=\mathrm{e}1\theta$(2. 7)${\rm Re}${ Zo$\mathrm{p}|$$(\mathrm{k}\geqq 1)$$=\mathrm{k}\mathrm{w}(\mathrm{Z}_{0})$[$(0\leqq\theta\leqq 2\pi)$

we see that,}$\mathrm{f}(\mathrm{z}_{0}))’$$(\mathrm{D}^{\iota 1\mathrm{t}|}\tau$( ト P)$=\mathrm{a}+{\rm Re}\{((\mathrm{p}-2\beta)\mathrm{e}\mathrm{i}\theta-\mathrm{p})/(1+\mathrm{e}\mathrm{i}\theta)\}+2(\mathrm{p}-\beta)\mathrm{k}$.

$\mathrm{n}$$+\mathrm{a}$$1{\rm Re}\{\mathrm{e}\mathrm{i}\mathit{0}(1+\mathrm{e}1\theta) 2\}$$=\mathrm{a}-\beta+(_{\backslash }\mathrm{p}-\beta)\mathrm{k}/((\mathrm{n}+\mathrm{p})(1\mathrm{t}\cos\theta)$$\leqq \mathrm{a}-\beta+(\mathrm{p}^{-}\beta)/2(\mathrm{n}\{\mathrm{p})=0$

for$\mathrm{a}>\mathrm{p}$and$\beta=$($\mathrm{p}$V2$\mathrm{a}(\mathrm{n}+\mathrm{p})$)$/(1+2(\mathrm{n}\dagger_{\mathrm{P})})$.

This contradicts our condition 2. $])$ . Therefore, lw(z)l $<1$ for all$($${\rm Re}\{z\mathrm{p}\}\uparrow(\mathrm{D}^{\mathrm{n}\}_{l}-\iota}’ \mathrm{f}(\mathrm{z}))’\}$

Next,$>-\beta$$(\mathrm{z}\epsilon \mathrm{U})$.

we prove

Theorem 2. Let(2. 8)$\mathrm{F}_{\mathrm{c}}(\mathrm{z})=\mathrm{c}\mathrm{z}^{-(}\mathrm{p}(\mathrm{z}\mathrm{O}\mathrm{t}^{(\}_{1)}}$\dag erf(t)dt$(\mathrm{c}>0)$$\mathrm{z}\epsilon \mathrm{U}$,

or

58

for

$\mathrm{f}(\mathrm{z})\in\Sigma_{l},$. If$\mathrm{f}(\mathrm{z})$(2. 9)

satisfies$>-\mathrm{a}$${\rm Re}\{\mathrm{z}^{\mathrm{p}}| (\mathrm{D}^{\mathrm{l}\mathrm{l}\mathrm{I}\mathrm{r}}’ 1\mathrm{f}(\mathrm{z}))’ \}$for some$\mathrm{a}(\mathrm{a}>\mathrm{p})$(2. 10)

where

then

${\rm Re}\{\mathrm{z}^{\mathrm{p}1} [ (\mathrm{D}^{\mathrm{r}11\mathrm{p}} [ 1_{\mathrm{c}}^{\mathrm{t}}\backslash (\mathrm{z}))^{r}\}>-\beta$$\beta=$$1^{\mathrm{J}}\mathrm{r}\alpha)\mathrm{f}$(2. 11)Then(2. 12),($\mathrm{p}+2$a )$\mathrm{c}$$/(1+2\mathrm{c})$,$(r, \epsilon \mathrm{u})$.

. We define the function$\mathrm{w}(\mathrm{z})$$\mathrm{z}^{\mathrm{p}11}(\mathrm{D}^{\mathrm{t}1}|\mathrm{p}| \mathrm{F}_{(}.

$\mathrm{w}(\mathrm{z})$$(\mathrm{z}\epsilon \mathrm{U})$is analytic in$\mathrm{U}$by(_{7_{\lrcorner}}))’=((\mathrm{p}-2\beta)\mathrm{w}(\mathrm{z})-\mathrm{p})/(1+\mathrm{w}(_{\mathrm{Z}},))$

and$\mathrm{w}(\mathrm{O})=0$$(\mathrm{w}(_{\mathrm{Z})\neq}-\iota)$. Noting that$\mathrm{z}(\mathrm{D}^{\mathrm{r}}1|\mathrm{f}’ 1 \mathrm{F}_{\mathrm{c}}(\mathrm{z}))’=\mathrm{c}\mathrm{D}^{11}|\mathrm{p}|\mathrm{f}(r)-(\mathrm{C}4_{\mathrm{P})}\mathrm{D}^{\mathrm{t})1\mathrm{p}}\mathrm{t}$$\Gamma_{\mathrm{c}}|$$(\mathrm{z})$,therefore we have(2. 13)$\mathrm{z}^{1\}}|1(\mathrm{D}^{\mathrm{n}\mathfrak{l}\mathrm{p}}1\mathrm{f}(\mathrm{z}))$’$=((\mathrm{p}-2\beta)\mathrm{w}(\mathrm{Z})-\mathrm{P})/(1+\mathrm{w}(\mathrm{z}))42(\mathrm{P}^{-}\beta)z\mathrm{w}’$$(\mathrm{z})/\mathrm{c}(1\{\mathrm{w}(_{\mathrm{Z}}))^{2})$Therefore, if we assume that there exists a point Zo$\max$

1lw(z)l $=1\mathrm{W}(\mathrm{Z}_{0})1=1$$\epsilon \mathrm{U}$${\rm Re}${ Zo$\mathrm{z}\mathrm{I}\leq..1\mathrm{z}_{\iota}1$$\mathrm{p}\vdash 1(\mathrm{D}^{l1\dagger}\mathrm{r}’|$$\leqq$$=0$$\mathrm{a}$$\mathrm{f}$($\mathrm{z}_{0}$) $)’$}

such that$(\mathrm{w}(\mathrm{z}_{\mathrm{t})})\neq-])$then Lemma gives us that
(2.14).

$+\mathrm{a}$$-\beta+(\mathrm{p}-\beta)/2\mathrm{c}$.

59which contradicts our condition (2. 9). This completes the$\mathrm{p}\mathrm{r}(\mathrm{x})\mathrm{f}$of Theorem 2.

References

N. E. Cho and M. Nullokawa, On certain subclass of meromorphical ly multivalentfunctions, Chinese J. Math. 22(1994), 197-202.
Math. Soc.
[2] I. S. Jack, Functions starlike and convex of order a, J.

[1]$1_{J}\mathrm{o}\mathrm{n}\mathrm{d}\mathrm{o}\mathrm{n}$3(1971),[3]

469-474.

S. S. Miller and P. T. Mocanu, Second order differential inequalities in thecomplex plane, J. Math. Anal. Appl. 65(1978), 289-305.
Department of Mathematics

Department of Mathemati cs

University of Gunma

University of Gunma

Maebashi, Gunma 371

Maebashi, Gunma 371

JapanJapan
Department of Mathematics

Gunma College of TechnologyToriba, Maebashi

Gunma 371
Japan