数理解析研究所講究録
第 1982 巻 2016 年 8-12

8On uniformly-type Sakaguchi functions
Junichi Nishiwaki and Shigeyoshi Owa
Abstract
be the class of analytic functions $f(z)$ in the open unit disc U. Furthermore,letconsisting of functions $f(z)$ related toandbe the subclasses of
uniformly convex and Sakaguchi functions. The object of the present paper is trying, and considering
to guess inclusive relations between uniformly convexity,and.
coeﬃcient inequalities for $f(z)$ belonging to the classLet$\mathcal{U}S_{s}$$\mathcal{A}$$\mathcal{A}$$\mathcal{U}S_{s}(\alpha, \beta)$$\mathcal{U}S_{s}$$S_{p}$$\mathcal{U}S_{s}(\alpha, \beta)$

1IntroductionLet$\mathcal{A}$be the class of functions $f(z)$ of the form$f(z)=z+ \sum_{n=2}^{\infty}a_{n}z^{n}$which are analytic in the open unit discto be starlike with respect to symmetrical points in. A function
if it satisfies$\mathbb{U}=\{z\in \mathbb{C} :$\mathbb{U}$|z|<1\}$${\rm Re}( \frac{zf’(z)}{f(z)-f(-z)})>0 (z\in \mathbb{U})$$f(z)\in \mathcal{A}$is said.

This class is introduced by Sakaguchi [7]. A function
is said to be in the classof uniformly convex (or starlike) functions denoted by
if $f(z)$ is convex (orstarlike) in and maps every circle or circular arc in with center at in onto the convex). These classes are introduced by Goodman [1]arc (or the starlike arc with respect to(see also [2]). For the class, it is defined as the one variable characterization by Rnning[5] and [6], that is, a function
is said to be in the classif it satisfies$f(z)\in \mathcal{A}$$\mathcal{U}C\mathcal{V}$$(or \mathcal{U}\mathcal{S}\mathcal{T})$$\mathbb{U}$$\mathbb{U}$$\zeta$$\mathbb{U}$$f(\zeta)$$\mathcal{U}C\mathcal{V}$$f(z)\in \mathcal{A}$$\mathcal{U}C\mathcal{V}$${\rm Re} \{1 十\frac{zf"(z)}{f’(z)}\}>|\frac{zf"(z)}{f’(z)}| (z\in \mathbb{U})$.

It is independently studied by Ma and Minda [3]. Further, a function
if it satisfies
the corresponding class denoted by$f(z)\in \mathcal{A}$is said to be$S_{p}$

2010 Mathematics Subject Classification: Primary $30C45$
Keywords and Phrases: Analytic function, Sakaguchi function, uniformly starlike,
uniformly convex, parabolic starlike.

9${\rm Re} \{\frac{zf’(z)}{f(z)}\}>|\frac{zf’(z)}{f(z)}-1| (z\in \mathbb{U})$.

This class
was introduced by nning [ . We easily know that the relation
if and only if $zf’(z)\in S_{p}$ . By virtue of these classes, we define the subclassconsisting of functions $f(z)$ which satisfy$5]$$R\emptyset$$\mathcal{S}_{p}$$f(z)\in \mathcal{U}C\mathcal{V}$$\mathcal{U}S_{s}(\alpha, \beta)$of$\mathcal{A}$${\rm Re} \{\frac{zf’(z)}{f(z)-f(-z)}\}>\alpha|\frac{zf’(z)}{f(z)-f(-z)}-\frac{1}{2}|+\beta (z\in \mathbb{U})$for some2$\alpha(\alpha\geqq 0)$

and$\beta(0\leqq\beta<\frac{1}{2})$. We denote$\mathcal{U}S_{s}(1,0)\equiv \mathcal{U}S_{s}.$Some examples of relation between the class
and$S_{p},$$\mathcal{U}C\mathcal{V}$$\mathcal{U}S_{s}$We don’t have the inclusion relation between the classandgive two examples to consider some relations between these classes.
$S_{p},$

Example 2.1. Let us consider the function$f(z)\in \mathcal{A}$$\mathcal{U}C\mathcal{V}$$\mathcal{U}S_{s}$as given by$f(z)=z+ \frac{1}{5}z^{3}.$Then we obtain${\rm Re}( \frac{zf’(z)}{f(z)-f(-z)})-|\frac{zf’(z)}{f(z)-f(-z)}-\frac{1}{2}| (z=re^{i\theta})$$= \frac{25+20r^{2}\cos 2\theta+3r^{4}}{50+20r^{2}\cos 2\theta+2r^{4}}-\frac{r^{2}}{\sqrt{25+10r^{2}\cos 2\theta+r^{4}}}$$\geqq\frac{(1-r)(5+2r)}{2(5-2r)}>0$

which shows that$f(z)\in \mathcal{U}S_{s}$. On the other hand, choosing$z= \frac{2}{3}e^{\frac{\pi}{2}i}$${\rm Re}(1+ \frac{zf"(z)}{f’(z)})-|\frac{zf"(z)}{f’(z)}|=-\frac{5}{11}$which shows that$f(z)\not\in \mathcal{U}C\mathcal{V}.$

Example 2.2. Let us consider the function$f(z)\in \mathcal{A}$as given by$f(z)=z+ \frac{1}{7}z^{4}.$Then we obtain${\rm Re}( \frac{zf’(z)}{f(z)})-|\frac{zf’(z)}{f(z)}-1| (z=re^{i\theta})$, we get. However, we10$= \frac{49+35r^{3}\cos 2\theta+4r^{6}}{49+14r^{3}\cos 2\theta+r^{6}}-\frac{3r^{3}}{\sqrt{49+14r^{3}\cos 2\theta+r^{6}}}$$\geqq\frac{7(1-r^{3})}{7-r^{3}}>0$

which shows that $f(z)\in S_{p}$ . On the other hand, choosing$z= \frac{23}{24}e^{\frac{\pi}{3}i}$, we get${\rm Re}( \frac{zf’(z)}{f(z)-f(-z)})-|\frac{zf’(z)}{f(z)-f(-z)}-\frac{1}{2}|=-\frac{71}{24192}$

which shows that3$f(z)\not\in \mathcal{U}S_{s}.$Coeﬃcient inequalities for the class$\mathcal{U}S_{s}(\alpha, \beta)$

Our aim of this section is to discuss some coeﬃcient inequalities for function $f(z)$ to bein the class.
$\mathcal{U}\mathcal{S}_{s}(\alpha, \beta)$

Theorem 3.1. If$f(z)\in \mathcal{A}$satisfies(3.1)$\sum_{n=2}^{\infty}[2(n-1)(1+\alpha)|a_{2n-2}|+\{2n-1-2\beta+2\alpha(n-1)\}|a_{2n-1}|]\leqq 1-2\beta$

for some$\alpha(\alpha\geqq 0)$

and$\beta(0\leqq\beta<\frac{1}{2})$, then$f(z)\in \mathcal{U}S_{s}(\alpha, \beta)$.

Let us consider an example for Theorem 3.1.
Example 3.1. supposing that the function$f(z)\in \mathcal{A}$as given by$f(z)=z+ \sum_{n=2}^{\infty}\frac{(1-2\beta)t\delta_{2n-2}}{2n(n-1)^{2}(1+\alpha)}z^{2n-2}+\sum_{n=2}^{\infty}\frac{(1-2\beta)(1-t)\delta_{2n-1}}{\{2n-1-2\beta+2\alpha(n-1)\}n(n-1)}z^{2n-1}$,for some(3.1)
yieldscoeficient$\alpha(\alpha\geqq 0)$$\beta(0\leqq\beta<\frac{1}{2})$, $t(0\leqq t\leqq 1)$ and $|\delta_{2n-2}|=|\delta_{2n-1}|=1$ . Then the$\sum_{n=2}^{\infty}[2(n-1)(1+\alpha)|a_{2n-2}|+\{2n-1-2\beta+2\alpha(n-1)\}|a_{2n-1}|]$$= \sum_{n=2}^{\infty}\{\frac{(1-2\beta)t}{n(n-1)}+\frac{(1-2\beta)(1-t)}{n(n-1)}\}$$\leqq 1-2\beta.$

This implies that$f(z)\in \mathcal{U}S_{s}(\alpha, \beta)$.

11
Theorem 3.2. If$f(z)\in \mathcal{U}S_{s}(\alpha, \beta)$, then$|a_{2}| \leqq\frac{1-2\beta}{|1-\alpha|},$$|a_{3}| \leqq\frac{1-2\beta}{|1-\alpha|},$$|a_{2n}| \leqq\frac{1-2\beta}{n|1-\alpha|}\prod_{j=1}^{n-1}(1+\frac{1-2\beta}{j|1-\alpha|}) (n=2,3,4, \cdots)$and$|a_{2n+1}| \leqq\frac{1-2\beta}{n|1-\alpha|}\prod_{j=1}^{n-1}(1+\frac{1-2\beta}{j|1-\alpha|}) (n=2,3,4, \cdots)$.

References[1] A. W. Goodman, On uniformly convex functions, Annal. Polon. Math. 56(1991), 87-92.
[2] A. W. Goodman, On uniformly starlike functions, J. Math. Anal. Appl. 155(1991), 364-370.
[3] W. Ma and D. Minda, Uniformly convex functions, Annal. Polon. Math. 57(1992), 165-175.
[4] S. Owa, T. Sekine, R. Yamakawa, Notes on Sakaguchi functions, Austral. J. Math. Appl.
3 (12) (2005), 1–7.
[5] F. Rnning, Uniformly convex functions and a corresponding class of starlike functions,
Proc. Amer. Math. Soc. 118(1993), 189–196.

[6] F.
nning, On uniform starlikeness and related properties of univalent functions, Complex Variables 24(1994), 233–239.
$R\emptyset$[7] K. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Japan 11(1959), 72–75.
[8] S. Shams, S. R. Kulkarni, and J. M. Jahangiri, Classes of uniformly starlike and convex
functions, Internat. J. Math. Math. Sci. 55(2004), 2959–2961.

Junichi Nishiwaki
Department of Mathematics and Physics
Setsunan University
Neyagawa, Osaka 572-8508 Japanemail:jerjun2002@yahoo. co.jp

12
Shigeyoshi Owa
Department of Mathematics
Faculty of Education
Yamato University Katayama 2-5-1, Suita, Osaka 564-0082
Japanemail:owa.shigeyoshi@yamato-u.ac.jp