数理解析研究所講究録
1062 巻 1998 年 120-127

120

Examples of Starlike Functions and Convex Functionsof order$\alpha$Tadayuki SEKINE*[関根忠行](日大薬学部)
Shigeyoshi OWA \dagger [尾和重義] (近畿大理工学部)
Rikuo YAMAKAWA++[山川陸夫] (芝浦工大工学部)
Abstract
H.Silverman determines certain coefficient inequalities and distortion theorems
andfor univalent functions with negative coefficients that are starlike of orderconvex of order . The same coefficient inequalities and the similar distortion theorems are obtained for such univalent functions with not always negative coefficients.
We give some examples of those univalent functions and illustrate the images of theexamples by Mathematica. Further we estimate those univalent functions.
$\alpha$$\alpha$1

Introduction
Let

$A$

denote the class of functions $f(z)$ of the form$f(z)=z+ \sum_{=n2}a_{k}Z^{k}\infty$(1)$(n\in \mathrm{N}=\{1,2,3, \cdots\})$that are analytic in the unit disk $U=\{z : |z|<1\}$ .
We first consider the so-called subclasses of analytic functions with negative coefficients. Let $A(n)$ denote the subclass of consisting of functions of the form
$A$$f(z)=z- \sum_{+k=n1}^{\infty}a_{k}z^{k}$$(a_{k}\geqq 0, n\in \mathrm{N}=\{1,2,3, \cdots\})$.

(2)

Let $T(n)$ denote the subclass of $A(n)$ consisting of functions which are univalent in U.
Further a function in $T(n)$ is said to be starlike of order $\alpha(0\leqq\alpha<1)$ if and only if itsatisfies${\rm Re} \{\frac{zf’(z)}{f(z)}\}>\alpha$$(z\in U)$(3)*College of Pharmacy, Nihon University, 7-7-1, Narashinodai, Funabashi, Chiba 274-8555, Japan\dagger Department of Mathematics, Kinki University, Higashi-Osaka, Osaka 577, Japan
Department of Mathematics, Shibaura Institute of Technology, 307, Shinsaku, Omiya-shi 330, Japan$\mathrm{t}$121

and such a subclass of $A(n)$ consisting of all the starlike functions of order is denoted. Also, $f(z)\in T(n)$ is said to be convex of order $\alpha(0\leqq\alpha<1)$ if and only if itbysatisfies$\alpha$$T_{\alpha}(n)$${\rm Re} \{1+\frac{zf^{J/}(_{\mathcal{Z}})}{f’(z)}\}>\alpha$(4)$(z\in U)$

and the subclass by. The classes $T(n),$andwere introduced byand these classes have been studied by Srivastava, Owa and,
Kiryakova, Saigo and Owa [2], and.
$C_{\alpha}(1)=C(\alpha)$ which
For $n=1$ , these notations are usually used aswere introduced by.
Using the same way ofdetermined a necessary and sufficients conditions for a function in $A(n)$ belongs to.
andTheorem. A function $f(z)$ in $A(n)$ is in
if and only if$C_{\alpha}(n)$$C_{\alpha}(n)$$T_{\alpha}(n)$$\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{j}\mathrm{e}\mathrm{a}[1]$$\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{j}\mathrm{e}\mathrm{a}[9]$$\mathrm{S}\mathrm{e}\mathrm{k}\mathrm{i}\mathrm{n}\mathrm{e}[4]$$T_{\alpha}(1)=T^{*}(\alpha),$$\mathrm{S}\mathrm{i}\mathrm{l}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{n}[8]$$\mathrm{S}\mathrm{i}\mathrm{l}\mathrm{V}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{n}[8],$$\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{j}\mathrm{e}\mathrm{a}[1]$$C_{\alpha}(n)$$T_{\alpha}(n)$$T_{\alpha}(n)$$\mathrm{A}(\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{j}\mathrm{e}\mathrm{a}[1])$$\sum_{k=n+1}^{\infty}(k-\alpha)a_{k}\leqq 1-\alpha$

Theorem$\mathrm{B}(\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{j}\mathrm{e}\mathrm{a}[1]).$$(0\leqq\alpha<1)$.

A function $f(z)$ in $A(n)$ is in$\sum_{k=n+1}^{\infty}k(k-\alpha)a_{k}\leqq 1-\alpha$(5)$C_{\alpha}(n)$$(0\leqq\alpha<1)$

if and only if.

(6)of , and the subclasses
Recently, in [5] we introduced the subclassandofin the following manner.
Letdenote the subclass of consisting of functions of the form
$A(n, \theta)$$C_{\alpha}(n, \theta)$$A$$T_{\alpha}^{*}(n, \theta)$$A(n, \theta)$$A$$A(n, \theta)$$f(z)=z- \sum_{k=n+1}^{\infty}e-1\theta a_{k^{Z}}i(k)k$$(a_{k}\geqq 0, n\in \mathrm{N})$(7).

We note that $A(n, 0)=A(n)$ , that is, $A(n, 0)$ is the subclass of analytic functions
the subclasses ofandwith negative coefficients. We denote byof starlike and convex functions of order in , respectively.
proved the same coefficients inequalities as Chatterjea showed for $f(z)$ in
as follows.
is in
Theorem (Sekine and Owa [5]). A function $f(z)$ inif andonly if$T_{\alpha}^{*}(n, \theta)$$\alpha$$A(n, \theta)$$C_{\alpha}(n, \theta)$$U$$A(n, \theta)$$\mathrm{t}\prime \mathrm{V}\mathrm{e}$$A(n, \theta)$$\mathrm{C}$$k=n+ \sum_{1}^{\infty}(k-\alpha)a_{k}\leqq 1-\alpha$

Theorem
only if$\mathrm{D}$$(0\leqq\alpha<1)$.

(8)(Sekine and Owa [5]). A function $f(z)$ in $A(n, \theta)$ is in$k=n+ \sum_{1}^{\infty}k(k-\alpha)a_{k}\leqq 1-\alpha$$T_{\alpha}^{*}(n, \theta)$$(0\leqq\alpha<1)$.

$C_{\alpha}(n, \theta)$if and(9)122Further we determined the following Distortion theorems.
Theorem (Sekine and Owa [5]). If $f(z)$ is in, then$\mathrm{E}$$T_{\alpha}^{*}(n, \theta)$$|z|- \frac{1-\alpha}{n+1-\alpha}|z|^{n+1}\leq|f(z)|\leq|z|+\frac{1-\alpha}{n+1-\alpha}|z|^{n+1}$.

(10)

The right-hand equality holds for the function$f(z)=z-e$

1$in\theta$$-\alpha$$\overline{n+1-\alpha}z^{n+1}$$(z=re^{-i(\theta+} \frac{\pi}{n}),$$r<1)$

(11)

.

(12)

and the left-hand equality holds for the function$f(z)=z-e$
Theorem$\mathrm{F}$$1-\alpha$$in\theta$$(z=re^{-i\theta}, r<1)$$\overline{n+1-\alpha}^{\mathcal{Z}^{n+1}}$(Sekine and Owa [5]). If $f(z)$ is in$C_{\alpha}(n, \theta)$, then$|z|- \frac{1-\alpha}{(n+1)(n+1-\alpha)}|z|^{n+1}\leq|f(z)|\leq|z|+\frac{1-\alpha}{(n+1)(n+1-\alpha)}|z|^{n+1}$

.

(13)

The right-hand equality holds for the function$f(z)=z-e^{in\theta_{\frac{1-\alpha}{(n+1)(n+1-\alpha)}z^{n+1}}}$$(z=re^{-i(} \theta+\frac{\pi}{n}),$$r<1)$

(14)

.

(15)

and the left-hand equality holds for the function$f(z)=z-e^{in\theta_{\frac{1-\alpha}{(n+1)(n+1-\alpha)}}1}z^{n+}$2$(z=re^{-i\theta}, r<1)$

ExamplesLet$A_{\alpha}(n, \theta, h)$denote the subclass of$A(n, \theta)$

consisting of functions of the form$f(z)=z- \sum_{k=n+1}^{\infty}ei(k-1)\theta kak,h^{\mathcal{Z}}$$(h\geqq-n)$ ,$(1-\alpha)^{2}$

where $a_{k,h}=\overline{(k+h-\alpha)(k+1+h-\alpha)(k-\alpha)}$Let$B_{\alpha}(n, \theta, h)$denote the subclass of$A(n, \theta)$$(0\leqq\alpha<1)$

Then we have the following theorems.

.

consisting of functions of the form$g(z)=z- \sum_{=kn+1}^{\infty}e^{i}b(k-1)\theta k,hz^{k}$where $b_{k,h}= \frac{(1-\alpha)^{2}}{(k+h-\alpha)(k+1+h-\alpha)(k-\alpha)k}$(16)$(h\geqq-n)$$(0\leqq\alpha<1)$

.

,

(17)

123

Theorem 2.1. If $f(z)\in A_{\alpha}(n, \theta, h)$ , then$f(z)\in T_{\alpha}^{*}(n, \theta)$.

Proof.
$\sum_{k=n+1}^{\infty}(k-\alpha)ak,h=\sum_{=kn+1}(k-\alpha)\infty\frac{(1-\alpha)^{2}}{(k+h-\alpha)(k+1+h-\alpha)(k-\alpha)}$$–(1- \alpha)2\sum_{+k=n1}^{\infty}(\frac{1}{k+h-\alpha}-\frac{1}{k+1+h-\alpha})$$=(1- \alpha)^{2}\frac{1}{n+1+h-\alpha}$$\frac{(1-\alpha)^{2}}{1-\alpha}=1-\alpha$$h=-n$ ,,$=\{$$\frac{(1-\alpha)^{2}}{n+1+h-\alpha}<\frac{(1-\alpha)^{2}}{1-\alpha}--1-\alpha$

Hence we know that $f(z)$ is an element ofUsing Theorem$\mathrm{B}$$h>-n$ .

by virtue of the theorem C.

$\square$, we can also prove the following theorem 2.2.

Theorem 2.2. If $g(z)\in B_{\alpha}(n, \theta, h)$ , thenIn the case of$T_{\alpha}^{*}(n, \theta)$,$\theta=0$

Example 2.1. If.

, these theorems were proved by Sekine and Yamanaka [6].

$f(z) \in A_{0}(1, \frac{\pi}{4}, -1)_{2}$$f(z)$$g(z)\in C_{\alpha}(n, \theta)$

then we have$=$$z- \sum_{k=2}e-k1)\frac{\pi}{4}\frac{1}{(k-1)k^{2}}i(Z\infty k$$=$$z- \frac{1+i}{4\sqrt{2}}z^{2}-\frac{i}{18}\mathcal{Z}^{3}+\frac{1-i}{48\sqrt{2}}z^{4}+\frac{1}{100}z^{5}$$+ \frac{1+?}{180\sqrt{2}}.z^{6}+\frac{i}{294}z^{7}-\frac{1-i}{448\sqrt{2}}Z8-\frac{1}{648}z-9\ldots$(18)

Example 2.2. If $g(z) \in B_{0}(1, \frac{\pi}{4}, -1)$ , then we have$g(z)$$=$$z- \sum_{2k=}^{\infty}e-k1)\frac{\pi}{4}\frac{1}{(k-1)k^{3}}i(Zk$$=$$z- \frac{1+i}{8\sqrt{2}}z^{2}-\frac{i}{54}Z+3\frac{1-i}{192\sqrt{2}}z^{4}+\frac{1}{500}z5$$+ \frac{1+i}{1080\sqrt{2}}z^{6}+\frac{i}{2058}Z^{\tau}-\frac{1-i}{3584\sqrt{2}}z8-\frac{1}{5832}z9-\cdots$(19)

We show the images of $|z|\leqq 1$ by the apporoximate expressions for the examples with
Mathematica. In view of the figures, we can image that the functions of the examples
2.1 and 2.2 are starlike function and convex function, respectively.

124

Figure 1: Image of$|z|\leqq 1$by

Figure 2: Image of$|z|\leqq 1$by $f(z)=z- \sum_{k2}^{9}=\frac{1}{(k-1)k3}e(k-1)\frac{\pi}{4}x^{k}i$$f(z)=z- \sum_{k2}^{9}=\frac{1}{(k-1)k^{2}}e^{i}(k-1)\frac{\pi}{4}xk$

125Further we estimate the functions inby the theorem we have$T_{\alpha}^{*}(n, \theta)$

and$C_{\alpha}(n, \theta)$. Let$f(z) \in T_{0}^{*}(1, \frac{\pi}{4})$, then$\mathrm{E}$$|z|- \frac{1}{2}|z|^{2}\leqq|f(z)|\leqq|z|+\frac{1}{2}|Z|^{2}$(20).

And the right-hand equality hold for the function $f(z)=z- \frac{1+i}{2\sqrt{2}}z^{2}$ on the harf line
. By letting $rarrow 1$ for the, also the left-hand equality on.
, we havefunction$z=re^{-i(\frac{\pi}{4})}$$z=re^{-(\pi+\frac{\pi}{4})}$$f(z)=z- \frac{1+i}{2\sqrt{2}}z^{2}$

Figure 3:$\frac{1}{2}\leqq|f(Z)|\leqq\frac{3}{2}$$\mathrm{I}\mathrm{m}\mathrm{a}_{\iota\supset}\sigma \mathrm{e}$ofFigure 4: Images of$|z|\leqq 1$

by$|z|\leqq 1$by$f(Z)=z- \frac{1+i}{2\sqrt{2}}z^{2}$$z- \frac{1+i}{2^{\sqrt{2}}}z^{2},$$\frac{z}{2}$and$\frac{3}{2}z$126

Also, in case of$g(z) \in C_{0}^{*}(1, \frac{\pi}{4})$, we have

Figure 5: image of

Figure 6: Images of$\frac{3}{4}\leqq|g(z)|\leqq\frac{5}{4}$$|z|\leqq 1$$|z|\leqq 1$by

by.

$g(z)=z- \frac{1+i}{4\sqrt{2}}z^{2}$$z- \frac{1+i}{4\sqrt{2}}z^{2},$$\frac{3}{4}z$

and$\frac{5}{4}z$

References[1] S. K. Chatterjea, On starlike functions, J. Pure Math. 1(1981), 23-26.

[2] V. S. Kiryakova, M. Saigo and S. Owa, Distortion and characterization theorems forstarlike and convex functions related to generalized fractional calculus, Publ. Res. Inst.
Math. Sci. 1012(1997), 25-46.

127[3] M. S. Robertson, On the therory of univalent functions, Ann. of Math. 37(1936), 374$- 408$ .
[4] T. Sekine, On new generalized classes of analytic functions with negative coefficients,
Report Res. Inst. Sci. Tec. Nihon Univ. 35(1987), 1-26.

[5] T. Sekine and S. Owa, New problems of coefficients inequalities, Publ. Res. Inst. Math.
Sci. 1012(1997), 164-176.
[6] T. Sekine and T. Yamanaka, Starlike functions and convex functions of order
negative coefficients, Math. Sci. Res. Hot-Line 1(1997), 7-12.

$\alpha$with[7] A. Schild and H. Silverman, Convolutions of univalent functions with negative coefficients, Ann. Univ. M. Curie-Sklodowska, Sect. A 12(1975), 99-106.
[8] H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc.
51(1975), 109-116.

[9] H. M. Srivastava, S. Owa and S. K. Chatterjea, A note on certainclasses of starlike
functions, Rend. Sem. Mat. Univ. Padova 77(1987), 115-124.