数理解析研究所講究録
第 1717 巻 2010 年 38-45

38-neighborhood defining bya new operator for certain analytic functions$(\alpha, \delta)$Kazuyuki Kugita , Kazuo Kuroki and Shigeyosi Owa
Abstract
For analytic functions $f(z)$ in the open unit disk , a new operator $D^{j}f(z)$ for anyinteger which is the generalization of Salagean differential operator and Alexander integral operator is introduced. The object of the present paper is to discuss some properties
for-neighborhood defining by a new operator $\dot{U}f(z)$ and to apply Miller-Mocaziulemma (J. Math. Anal. Appl. 65(1978)) for-neighborhood.
$U$

$j$

$(\alpha, \delta)$$(\alpha_{:}\delta)$

1

Introduction and definitions
Let$\mathcal{A}$be the class of functions $f(z)$ of the form$f(z)=z+ \sum_{n=2}^{\infty}a_{n}z^{n}$that are analytic in the open unit disk $U=\{z\in \mathbb{C} : |z|<1\}$ . For
introduced the following operator $D^{j}f(z)$ which is called Salagean differential operator.

$f(z)\in \mathcal{A}_{i}S\check{a}l\check{a}gean[3]$has,$D^{0}f(z)=f(z)=z+ \sum_{n=2}^{\infty}a_{\eta}z^{n}$$D^{1}f(z)=Df(z)=zf’(z)=z+ \sum_{n=2}^{\infty}na_{n}z^{n}$and$\dot{U}f(z)=D(\dot{p}^{-1}f(z))=z+\sum_{n=2}^{oc}n^{j}a_{n}z^{n}$$(j=1,2,3, \cdots)$.

Also, Alexander [1] has defined the following Alexander integral operator$D^{-1}f(z)= \int_{0}^{z}\frac{f(\zeta)}{(}d\zeta=z+\sum_{n=2}^{\infty}n^{-1}a_{n}z^{n}$.

2000 Mathematics Subject Classification: Primary $30C45$ .
Key Words and Phrases: Salagean differential operator, Alexader integral operator, neighborhood, Miller-Mocanu lemma.

39Futher, we introduce$(j=1,2,3, \cdots)$$D^{-j}f(z)=D^{-1}(D^{-(j-1)}f(z))=z+ \sum_{n=2}^{\infty}n^{-j}a_{n}z^{n}$which is the generalization integral operator of Alexander integral operator. Therefore, combining Salagean differential operator and Alexander integral operator, we introduce the operator$D^{j}f(z)$ by$Uf(z)=z+ \sum_{n=2}^{x}n^{j}a_{n}z^{n}$for any integer . Applying the above operator, we consider the subclassof $A$ as follows. A function $f(z)\in A$ is said to be in the class
if it satisfies$(\alpha_{1}, \alpha_{2}, \cdots :$j$$N_{m+1}^{j|1}(g_{1},g_{2}, \cdots, g_{p})$\alpha_{p};\delta)-$$(\alpha_{1}, \alpha_{2}, \cdots, \alpha_{p};\delta)-$$N_{m+1}^{j+1}(g_{1},g_{2}, \cdots, g_{p})$$(z\in U)$$| \frac{D^{j+1}f(z)}{z}-\sum_{k=1}^{p}\alpha_{k}\frac{D^{m+1}g_{k}(z)}{z}|<\delta$

for some$\delta>\sqrt{1-2\sum_{k--1}^{p}|\alpha_{k}|\cos d’+(\sum_{k--1}^{p}|\alpha_{k}|)^{2}}$, where $\beta=\arg\alpha_{k}$ for all$(k=1,2, \cdots,p)$$N_{m1}^{j^{\frac{\leq}{+1+}}}(g_{1}, g_{2},\cdot\cdot,g_{p})by\beta\pi,and.forsomeg_{k}(z)\in A$.

Let us define$k$with $-\pi\leqq$$(\alpha_{1}, \alpha_{2}, \cdots, \alpha_{p};\delta)-$$(\alpha, \delta)-N_{m+1}^{j+1}(g)\equiv(\alpha_{1}, \alpha_{2}\alpha_{p};\delta)-N_{m+1}^{j+1}(g_{1},g_{2}, \cdots,g_{p})$through this paper.

2Main theorem
Let us define $g_{A}.(z)\in A$$(k=1,2, \cdots,p)$ by$g_{k}(z)=z+ \sum_{n=2}^{\infty}b_{n,k^{Z^{n}}}$through this paper. Our first result of $f(z)$ for

Theorem 2.1$(\alpha, \delta)-N_{m+1}^{j+1}(g)$is contained inIf $f(z)\in A$ satisfies$\sum_{n=2}^{\infty}n|n^{j}a_{n}-n^{m}\sum_{k\cdot=1}^{p}\alpha_{k}b_{n,k}|\leqq\delta-\sqrt{1-2\sum_{k--1}^{p}|\alpha_{k}|\cos\beta+(\sum_{k--1}^{p}|\alpha_{k}|)^{2}}$for some$\delta>\sqrt{1-2\sum_{k--1}^{p}|\alpha_{k}|\cos_{f}f+(\sum_{k--1}^{p}|\alpha_{k}|)^{2}}$and for some $g_{k}(z)\in A$, where$’\prime i=\arg\alpha_{k}$for all $kwith-\pi\leqq\beta\leqq\pi$ ,$(k=1,2, \cdots,p)$ , then $f(z)\in(\alpha, \delta)-N_{m+1}^{j+1}(g)$ .

40
Proof.

Note that$|_{\approx}^{\underline{U^{+1}f(z)}}- \sum_{k=1}^{p}\alpha_{k}\frac{D^{m+1}g_{k}(z)}{z}|=|1+\sum_{n=2}^{\infty}n^{j+1}a_{n}z^{n-1}-\sum_{k=1}^{p}\alpha_{k}(1+\sum_{n=2}^{\infty}n^{m+1}b_{n,k}z^{n-1})|$$=|1- \sum_{k=1}^{p}\alpha_{k}+\sum_{n=2}^{\infty}n(n^{j}a_{n}-\uparrow\tau^{m}\sum_{k=1}^{p}\alpha_{k}b_{n,k})\sim\sim^{n-11}$$\leqq|1-\sum_{k=1}^{p}\alpha_{k}|+\sum_{n=2}^{\infty}n|n^{j}a_{n}-n^{m}\sum_{k=1}^{p}\alpha_{k}b_{n,k}||z|^{n-1}$$< \sqrt{1-2\sum_{\kappa--1}^{p}|\alpha_{k}|\cos\beta+(\sum_{k--1}^{p}|\alpha_{k}|)^{2}}+\sum_{n=2}^{\infty}n|n^{j}a_{n}-n^{m}\sum_{k=1}^{p}\alpha_{k}b_{n,k}|$If,$\sum_{n=2}^{\infty}n|n^{j}a_{n}-n^{m}\sum_{k=1}^{p}\alpha_{k}b_{n,k}|\leqq\delta-\sqrt{1-2\sum_{k--1}^{p}|\alpha_{k}|\cos 3+(\sum_{k--1}^{p}|\alpha_{k}|)^{2}}$then we see that$| \frac{U^{+1}f(z)}{z}-\sum_{k=1}^{p}\alpha_{k}\frac{D^{m+1}g_{k}(z)}{z}|<\delta$$(z\in U)$.

This gives us that $f(z)\in(\alpha, \delta)-N_{m+1}^{j+1}(g)$ .
Example 2.2$\square$For given $g_{k}(z)=z+ \sum_{n=2}^{\infty}b_{n,k}z^{n}\in A$ , we consider $f(z)=z+ \sum_{n=2}^{\infty}a_{n}z^{n}\in A$ with$a_{n}= \frac{\delta-\sqrt{1-2\sum_{k--1}^{p}|\alpha_{k}|\cos\beta+(\sum_{k--1}^{p}|\alpha_{k}|)^{2}}}{r\nu+2(n-1)}e^{i\gamma}+n^{m-j}\sum_{k=1}^{p}\alpha_{k}b_{n,k}$$(n=2,3,4, \cdots)$

Then, we have that$\sum_{n=2}^{\infty}n|n^{j}a_{n}-n^{m}\sum_{k=1}^{p}\alpha_{k}b_{n,k}|=\sum_{n=2}^{x}n|n^{j}\frac{\delta-\sqrt{1-2\sum_{k--1}^{p}|\alpha_{k}|\cos\beta+(\sum_{k--1}^{p}|\alpha_{k}|)^{2}}}{\dot{M}^{+2}(n-1)}e^{i\gamma}|$.

$=\delta-\sqrt{1-2\sum_{k--1}^{p}|\alpha_{k}|\cos\beta+(\sum_{k--1}^{p}|\alpha_{k}|)^{2}}$Therefore, $f(z)\in(\alpha, \delta)-N_{m+1}^{j+1}(g)$ .

.

.

41
In view of Theorem 2.1, we have the following corollary.
Corollary 2.3 Let $f(z)\in A$ satisfy$\sum_{n=2}^{\infty}n|n^{j}|a_{n}|-n^{m}\sum_{k=1}^{p}|\alpha_{k}||b_{n,k}||\leqq\delta-\sqrt{1-2\sum_{k--1}^{p}|\alpha_{k}|\cos\beta+(\sum_{k--1}^{p}|\alpha_{k}|)^{2}}$for some

for all $kwith-\pi\leqq\beta\leqq\pi$ ,$(n=2,3,4, \cdots)$ for all$(k=1,2, \cdots,p)$ with $\arg a_{n}-\arg b_{n,k}=\beta$, where$\delta>\sqrt{1-2\sum_{k--1}^{p}|\alpha_{k}|\cos_{f}f+(\sum_{k--1}^{p}|\alpha_{k}|)^{2}}$$\beta=\arg\alpha_{k}$and for some $g_{k}(z)\in A$, then $f(z)\in(\alpha, \delta)-N_{m+1}^{j+1}(g)$ .

$k$

Proof.

By Theorem 2.1, we have that if $f(z)\in A$ satisfies$\sum_{n=2}^{\infty}n|n^{j}a_{n}-n^{m}\sum_{k=1}^{p}\alpha_{k}b_{n,k}|\leqq\delta-\sqrt{1-2\sum_{k--1}^{p}|\alpha_{k}|\cos\beta+(\sum_{k--1}^{p}|\alpha_{k}|)^{2}}$then $f(z)\in(\alpha, \delta)-N_{m+1}^{j+1}(g)$ . Since $\arg a_{n}-\arg b_{n,k}=\beta$ , if
Therefore, we see that$\arg a_{n}=\varphi_{n}$, then,$\arg b_{n,k}=\varphi_{n}-\beta$.

$n^{j}a_{n}-n^{m} \sum_{k=1}^{p}\alpha_{k}b_{n,k}=n^{j}|a_{n}|e^{i\varphi_{n}}-n^{m}\sum_{k=1}^{p}|\alpha_{k}|e^{i\beta}|b_{n,k}|e^{i(\varphi_{n}-\beta)}$$=(n^{j}|a_{n}|-n^{m} \sum_{k=1}^{p}|\alpha_{k}||b_{n,k}|)e^{;_{\varphi_{n}}}$,that is, that$|n^{j}a_{n}-n^{m} \sum_{k=1}^{p}\alpha_{k}b_{n,k}|=|n^{j}|a_{n}|-n^{m}\sum_{k=1}^{p}|\alpha_{k}||b_{n,k}||$.

This completes the proof of the corollary.

$\square$Next, we discuss the necessary conditions for neighborhoods.

Theorem 2.4

If$f(z)\in(\alpha, \delta)-N_{m+1}^{j+1}(g)$with$\arg(n^{j}a_{n}-n^{m}\sum_{k=1}^{p}\alpha_{k}b_{n,k})=(n-1)\varphi$

for $n=2,3,4,$$\cdots$$(\varphi\in \mathbb{R})$,, then,.

$\sum_{n=2}^{\infty}n|n^{j}a_{n}-n^{m}\sum_{k=1}^{p}\alpha_{k}b_{n,k}|\leqq-1+\sum_{k=1}^{p}|\alpha_{k}|\cos\beta+\sqrt{\delta^{2}-(\sum_{k--1}^{p}|\alpha_{k}|\sin\beta)^{2}}$42Proof.

For $f(z)\in(\alpha, \delta)-N_{m+1}^{j+1}(g)$ , if we consider a point $z\in U$ such that $\arg z=-\ell’$ , then$z^{n-1}=|\approx|^{n-1}e^{-i(n-1)\varphi}$,and hence we have$| \frac{D^{j+1}f(z)}{z}-\sum_{k=1}^{p}\alpha_{k}\frac{D^{m+1}g_{k}(z)}{z}|=|1-\sum_{k=1}^{p}\alpha_{k}+\sum_{\mathfrak{n}=2}^{\infty}n(n^{j}a_{n}-n^{m}\sum_{k=1}^{p}\alpha_{k}b_{n,k})z^{n-1}|$.

$=|1- \sum_{k=1}^{p}\alpha_{k}+\sum_{n=2}^{\infty}n|n^{j}a_{n}-n^{m}\sum_{k=1}^{p}\alpha_{k}b_{n,k}||z|^{n-1}|<\delta$

Letting $|z|arrow 1^{-}$ we have$|k|$$= \{(1-\sum_{k=1}^{p}|\alpha_{k}|\cos\beta+\sum_{n=2}^{\infty}n|n^{j}a_{n}-n^{m}\sum_{k=1}^{p}\alpha_{k}b_{n,k}|)^{2}+(\sum_{k=1}^{p}|\alpha_{k}|\sin\beta)^{2}\}^{:}\leqq\delta$

which implies that$( \sum_{n=2}^{\infty}n|n^{j}a_{n}-n^{m}\sum_{k=1}^{p}\alpha kb_{n,k}|)^{2}+2(1-\sum_{k=1}^{p}|\alpha_{k}|\cos\beta)(\sum_{n=2}^{\infty}n|n^{j}a_{n}-n^{m}\sum_{k=1}^{p}\alpha_{k}b_{n,k}|)$.

$+1+( \sum_{k=1}^{p}|\alpha_{k}|)^{2}-2\sum_{k=1}^{p}|\alpha_{k}|\cos\beta-\delta^{2}\leqq 0$

Therefore, it is easy to see that.

$\sum_{n=2}^{\infty}n|n^{j}a_{n}-n^{m}\sum_{k=1}^{p}\alpha_{k}b_{n,k}|\leqq-1+\sum_{k=1}^{p}|\alpha_{k}|\cos\beta+\sqrt{\delta^{2}-(\sum_{k--1}^{p}|\alpha_{k}|\sin\beta)^{2}}$$\square$3

Applications of Miller-Mocanu lemma

In this section, we will give a certain implication for the classour probrem, we need the following lemma given by Miller and Mocanu [2].

$(\alpha, \delta)-N_{m+1}^{j+1}(g)$. To considering

be a positive integer, and let $F(z)$ be analytic in with $F^{(k)}(0)=0$ $(k=$$1,2,$$n-1),$ $F(O)=a$ andfor a complex number . If there exists a point $z_{0}\in U$
such that

Lemma 3.1$\cdots,$Let

$U$

$n$

$F(z)\not\equiv a$$a$$\max|F(z)|=|F(z_{0})|$ ,$|z|\leqq|zo|$then$\frac{z_{0}F’(\approx 0)}{F(z_{0})}=m$,,43where $m$ is real and$m \geqq n\frac{|F(z_{0})-a|^{2}}{|F(z_{0})|^{2}-|a|^{2}}\geqq n\frac{|F(\vee)|-|a|}{|F(z_{0})|+|a|}$.

Applying Lemma 3.1, we derive
Theorem 3.2

If $f(z)\in A$ satisfies$(z\in U)$$| \frac{D^{j+1}f(z)}{z}-\sum_{k=1}^{p}\alpha_{k}\frac{D^{m+1}g_{k}(z)}{z}|<\frac{2\grave{\delta}^{2}}{\delta+\sqrt{1-2\sum_{k--1}^{p}|\alpha_{k}|\cos\beta+(\sum_{k=1}^{p}|\alpha_{k}|)^{2}}}$for some
and for some

, where$\delta>\sqrt{1-2\sum_{k--1}^{p}|\alpha_{k}|\cos_{t}f+(\sum_{k--1}^{p}|\alpha_{k}|)^{2}}$$g_{k}(z)\in \mathcal{A}$$(k=1,2, \cdots ,p)$ ,for all$\prime f=\arg\alpha_{k}$$kwith-\pi\leqq\prime^{j}’\leqq\pi$

,

then

$| \frac{D^{j}f(\approx)}{z}-\sum_{k=1}^{p}\alpha_{k}\frac{D^{m}g_{k}(z)}{z}|<\delta$$(\sim\vee\in U)$,

which implies that $f(z)\in(\alpha, \delta)-N_{m+1}^{j+1}(g)$ .

Proof.

We define the function $F(z)$ by$F(z)= \frac{D^{j}f(z)}{z}-\sum_{k=1}^{p}\alpha_{k}\frac{D^{m}g_{k}(z)}{z}$$(z\in U)$.

Then,$\frac{zF’(z)}{F(z)}=\frac{\frac{D^{j+1}f(z)}{z}-\frac{D^{j}f(z)}{z}-\sum_{k=1}^{p}\alpha_{k}(\frac{D^{m+1}g_{k}(z)}{z}-\frac{D^{m}g_{k}(\approx)}{z})}{\frac{D^{j}f(z)}{z}-\sum_{k=1}^{p}\alpha_{k^{\frac{D^{m}g_{k}(z)}{z}}}}$$= \frac{1}{F(z)}(\frac{D^{j+1}f(z)}{z}-\sum_{k=1}^{p}\alpha_{k}\frac{D^{m+1}g_{k}(z)}{z})-1$.

Therefore,$| \frac{D^{j+1}f(z)}{z}-\sum_{k=1}^{p}\alpha_{k}\frac{D^{m+1}g_{k}(z)}{z}|=(1+\frac{zF’(z)}{F(z)})F(z)$Then $F(z)$ is analytic inwith$U$

us suppose that there is a point$F( O)=1-\sum_{k=1}^{p}\alpha_{k}$$z_{0}\in U$.

and $|F(0)|<\delta$ . In view of the condition, letsuch that $\max_{|z|\leqq|z_{0}|}|F(z)|=|F(z_{0})|=\delta$ . Then, by Lemma

3.1, we can write that

$F(z_{0})=\delta e^{i\theta}$,$\frac{\sim\gamma 0F’(\vee)}{F(z_{0})}=m$

and$m \geqq\frac{|\delta e^{i0}-(1-\sum_{k--1}^{p}\alpha_{k})|^{2}}{\delta^{2}-|1-\sum_{k=1}^{p}\alpha_{k}|^{2}}$.

44
Therefore, we see that$| \frac{D^{j+1}f(z_{0})}{z_{0}}-\sum_{k=1}^{p}\alpha_{k_{\gamma}^{\frac{D^{m+1}g_{k}(_{\sim 0}\gamma)}{\sim 0}1}}=|1+m||F(z_{0})|$$=\delta(1+m)$$\geqq\delta+\delta\frac{|\delta e^{i\theta}-(1-\sum_{k--1}^{p}\alpha_{k})|^{2}}{\delta^{2}-|1-\sum_{k=1}^{p}\alpha_{k}|^{2}}$$\geqq\delta+\delta\frac{\delta-|1-\sum_{k=1}^{p}\alpha_{k}|}{\delta+|1-\sum_{k=1}^{p}\alpha_{k}|}$.
$= \frac{2\tilde{\delta}^{2}}{\delta+\sqrt{1-2\sum_{k--1}^{p}|\alpha_{k}|\cos\beta+(\sum_{k--1}^{p}|\alpha_{k}|)^{2}}}$This contradicts our condition in Theorem 3.2. Thus, there is no point $z_{0}\in U$ such that$|F(z_{0})|=\delta$ . This means that $|F(z)|<\delta$ for all $z\in$ U. Therefore, we have that$(z\in U)$$| \frac{pf(z)}{z}-\sum_{k=1}^{p}\alpha k^{\frac{D^{m}g_{k}(z)}{z}1}<\delta$.
$\square$Taking $p=1$ in Theorem 3.2, and letting

and$\alpha_{1}=e^{i\alpha}$$g_{1}(z)=g(z)$,we find the following corollary.
Corollary 3.3

If $f(z)\in A$ satisfies$(z\in U)$$| \frac{D^{j+1}f(z)}{z}-e^{ia}\frac{D^{m+1}g(z)}{z}|<\frac{2\delta^{2}}{\delta+\sqrt{2(1-\cos\alpha)}}$for$some-\pi\leqq\alpha\leqq\pi,$or and for some$\delta>\sqrt{2(1-\cos}$$)$$| \frac{Uf(z)}{z}-e^{i\alpha}\frac{D^{m}g(z)}{z}|<\delta$

In paticular, by putting
obtain the assertion as follows.

$\delta=\tilde{\delta}+\sqrt{2(1-\cos\alpha)}$Corollary 3.4(3.1)$g(z)\in \mathcal{A}$, then$(z\in U)$.

for some $-\pi\leqq$ a$\leqq\pi$and$\tilde{\delta}>0$

If $f(z)\in A$ satisfies$| \frac{D^{j+1}f(z)}{z}-e^{i\alpha}\frac{D^{m+1}g(z)}{z}|<2\tilde{\delta}+\frac{4(1-\cos\alpha)}{\tilde{\delta}+2\sqrt{2(1-\cos\alpha)}}$$(z\in U)$, we can45
for$some-\pi\leqq\alpha\leqq\pi,\tilde{\delta}>0$(3.2)

and for some $g(z)\in A$ , then$| \frac{\dot{D}f(z)}{z}-e^{i\alpha}\frac{D^{m}g(z)}{z}|<\overline{\delta}+\sqrt{2(1-\cos\alpha)}$$(z\in U)$ .

Remark 3.5 Recently, in the paper by Kugita, Kuroki and Owa [4], we obtained the implication that
(3.3)

$|$

$f_{z}^{\dot{fl}^{+1}f(z)}-e^{i\alpha} \frac{D^{m+1}g(z)}{z}|<2\tilde{\delta}-\sqrt{2(1-\cos\alpha)}$$(z\in U)$

implies the inequality (3.2), where. Here, a simple check gives us that ifsatisfies the inequality (3.3), then $f(z)$ satisfies the inequality (3.1). Hence, it follows
this fact that if $f(z)\in A$ satisfies the assertion of Corollary 3.4, then the implication whichwere proven by Kugita, Kuroki and Owa [4] holds.
$\overline{\delta}>\sqrt{2(1-\cos\alpha)}$$f(z)\in \mathcal{A}$

References[1] J. W. Alexander, Functions which map the interior of the unit circle upon simple regions,Ann. of Math. 17(1915), 12-22.
[2] S. S. Miller and P. T. Mocanu, Second order differential inequalities in the complex plane,
J. Math. Anal. Appl. 65(1978), 289-305.

[3] G. S.

$S\check{a}l\check{a}gean$372.

, Subclass of univalent functions, Lecture Notes in Math. 1013(1983), 362-

[4] K. Kugita, K. Kuroki and S. Owa,
-neighborhood for functions associated withoperator
and Alexander integral operator, Int. J. Math. Anal. 4(2010),
differential$(\alpha, \delta)$$S\check{a}l\check{a}gean$

211-220.

Department of Mathematics
Kinki University
Higashi-Osaka, Osaka 577-8502
JapanE-mail: ib3mi3@bma.biglobe.ne.jp
freedom@sakai.zaq.ne.jp
owa@math.kindai.ac.jp