Owa, Shigeyoshi

Sekine, Tadayuki

Yamakawa, Rikuo

Kyoto University Research Information Repository

TitleNotes on Sakaguchi functions (Coefficient Inequalities inUnivalent Function Theory and Related Topics)Author(s)Owa, Shigeyoshi; Sekine, Tadayuki; Yamakawa, RikuoCitation数理解析研究所講究録 (2005), 1414: 76-82Issue Date2005-02URL http://hdl.handle.net/2433/26232RightType Departmental Bulletin PaperTextversionpublisherKyoto University76Notes on Sakaguchi functionsShigeyoshi Owa, Tadayuki Sekine and Rikuo YamakawaAbstractBy using the definition for certain univalent functions $f(z)$ in the open unit disk $\mathrm{U}$given by K.Sakaguchi(J.Math.Soc.Japan, 11(1959)), two classes $S(\alpha)$ and $\mathcal{T}(\alpha)$ of analyticfunctions in $\mathrm{U}$ are introduced. The object of the present paper is to discuss some propertiesof functions $f(z)$ belonging to the classes $S(\alpha)$ and $\mathcal{T}(\alpha)$ .1IntroductionLet $A$ be the class of functions of the form$f(\approx)=z$ $+ \sum_{n=2}^{\infty}a_{n}z^{n}$ (1.1)that are analytic in the open unit disk $\mathrm{U}=\{z\in \mathbb{C}||z| <1\}$ . Afunction $f(z)\in A$ is saidto be in the class $S(\alpha)$ if it satisfies${\rm Re} \{\frac{zf’(z)}{f(z)-f(-z)}\}>$ (1.2)for some $\alpha(0\leqq\alpha<\frac{1}{2})$ and for all $z$ $\in \mathrm{u}$. The class $S(0)$ when $\alpha=0$ was introducedby Sakaguchi [2]. Therefore, afunction $f(z)\in S(\alpha)$ is called Sakaguchi function of order$\alpha$ . We also denote by $\mathcal{T}(\alpha)$ the subclass of $A$ consisting of all functions $f(z)$ such that$\approx f’(_{\sim}’)\in S(\alpha)$ .For $f(z)$ belonging to $S(\alpha)$ and $\mathcal{T}(\alpha)$ , Cho, Kwon and Owa [1] have givenLen ma 1If $f(z)\in A$ satisfies$\sum_{n=2}^{\infty}\{2(n-1)|a_{2n-2}|+(2n-1-2\alpha)|a_{2n-1}|\}\leqq 1-2\alpha$ (1.3)for some $\alpha(0\leqq\alpha<\frac{1}{2})$ , then $f(z)\in S(\alpha)$ .Lemma 2If $f(z)\in A$ satisfies2004 Mathematics Subject Classification: Primary $30\mathrm{C}45$ .Ken Words and Phrases Sakaguchi function, coefficient inequality, distortion inequality1414 2005 76-8277$\sum_{n=2}^{\infty}\{4(n-1)^{2}|a_{2n-2}|+(2n-1)(2n-1-2\alpha)|a_{2n-1}|\}\leqq 1-2\alpha$ (1.4)for some $\alpha(0\leqq\alpha<\frac{1}{2})$ , then $f(z)\in \mathcal{T}(\alpha)$ .In view of the abobe lemmas, we seeExample 1.1 Let us consider a function $f(z)$ given by$f(z)=z+ \frac{1}{3}\delta_{2}z^{2}+(1-\frac{8}{3(3-2\alpha)})\delta_{3}z^{3}$ (1.5)with $|\delta_{2}|=|\delta_{3}|=1$ . Then, since$\sum_{n=2}^{\infty}\{2(n-1)|a_{2\mathrm{n}-2}|+(2n-1-2\alpha)|a_{2n-1}|\}<1-2\alpha$we see that $f(z)\in S(\alpha)$ .Example 1.2 Let us consider a function $f(z)$ given by$f(z)=z$ $+ \frac{1}{6}\delta_{2}z^{2}+\frac{1}{3}(1-\frac{8}{3(3-2\alpha)})\delta_{3}z^{3}$ (1.6)with $|\delta_{2}|=|\delta_{3}|=1$ . Then, since$zf’(z)=z+ \frac{1}{3}\delta_{2}z^{2}+(1-\frac{8}{3(3-2\alpha)})\delta_{3}z^{3}\in S(\alpha)$ ,we have that $f(z)\in \mathcal{T}(\alpha)$ .2 Coefficient inequalitiesApplying Carath\’eodory function$p(z)=1+ \sum_{n=1}^{\infty}p_{n}z^{n}$ (2.1)in $\mathrm{u}$ , we first discuss the coefficient inequalities for functions $f(z)$ in $S(\alpha)$ and $\mathcal{T}(\alpha)$ .Theorem 2.1 If $f(z)\in S(\alpha)$ , then$|a_{2n}| \leqq\frac{\prod_{j=1}^{n+1}(j-2\alpha)}{n(n!)}$ $(n\geqq 1)$ (2.2)and$|a_{2n+1}| \leqq\frac{\prod_{j=1}^{n}(j-2\alpha)}{n!}$ $(n\geqq 1)$ . (2.3)78Proof 1Ve define the function $p(z)$ by$p(z)= \frac{1}{1-2\alpha}(\frac{2zf’(z)}{f(z)-f(-z)}-2\alpha)=1+\sum_{n=1}^{\infty}p_{n^{\mathcal{Z}^{\eta}}}$ (2.4)for $f(z)\in S(\alpha)$ . Then $p(z)$ is a Caratheodory function and satisfies $|p_{n}|\leqq 2(n\geqq 1)$ .Since$2zf’(z)=(f(z)-f(-z))((1-2\alpha)p(z)+2\alpha)$ ,we obtain that$a_{2n}= \frac{1-2\alpha}{2n}(p_{2n+1}+a_{3}p_{2n-1}+\cdots+a_{2n+1}p_{1})$ (2.5)and$a_{2\mathfrak{n}+1}= \frac{1-\sim 9\alpha}{2n}$ $(p_{2n}+a_{3}p_{2n-2}+\cdots +o_{2n-1}p_{2})$ . (2.6)Taking $n=1$ , we see that$|a_{3}|\leqq 1-2\alpha$ (2.7)and$|a_{2}|= \frac{1-\underline{9}\alpha}{1+|a_{3}|}\leqq(1-2\alpha)(2-2\alpha)$ . (2.8)Thus, using the mathematical induction, we complete the proof of the theorem.Remark 2.1 Equalities in Theorem 2.1 are attended for $f(z)$given by$\frac{zf_{(}^{\prime/}z)}{f(z)-f(-z)}=\frac{1+(1-4\alpha)z}{2(1-z)}$ .Theorem 2.2 If $f(z)\in \mathcal{T}(\alpha)$ , then$|a_{2n}| \leqq\frac{\prod_{j=1}^{n+1}(j-2\alpha)}{2n^{2}(n!)}$ $(n\geqq 1)$ (2.9)and$|a_{2n+1}| \leqq\frac{\prod_{j=1}^{n}(j-2\alpha)}{(2n+1)(n!)}$ $(n \geqq 1)$‘ (2.’10)3 Distortion inequalit\’ies73In view of Lemma 1 and Lemma 2, we introduce the subclasses $S_{0}(\alpha)$ and $\mathrm{T}(\mathrm{a})$ . If$f(z)\in S(\alpha)$ satisfies the coefficient inequalities (1.3), then we say that $f(z)\in \mathrm{S}0(\mathrm{a})$ . Also,if $f(z)\in \mathcal{T}(\alpha)$ satisfies the coefficient inequalities (1.4), then we say that $f(z)\in \mathrm{T}(\mathrm{a})$ .For $f(z)$ belonging to $S_{0}(\alpha)$ and $\mathcal{T}_{0}(\alpha)_{\backslash }$ Cho, $\mathrm{I}\backslash ^{\nearrow}\mathrm{w}\mathrm{o}\mathrm{n}$ and Owa [1] have shown thatTheorem A If $f(z)\in \mathrm{S}0(\mathrm{a})f$ then$|z|- \frac{1-2\alpha}{\underline{9}}|z|^{2}-\frac{1-2\alpha}{3-\sim 9\alpha}|z|^{3}\leqq|f(z)|\leqq|z|+\frac{1-2\alpha}{2}|z|^{2}+\frac{1-2\alpha}{3-2\alpha}|z|^{3}$ (3.1)and$1-(1-2 \alpha)|z|-\frac{3(1-2\alpha)}{3-2\alpha}|z|^{2}\leqq|f’(z)|\leqq 1+(1-2\alpha)|z|+\frac{3(1-2\alpha)}{3-2\alpha}|z|^{2}$ (3.2)for $\overline{\underline{/}}\in \mathrm{U}$ .Theorem $\mathrm{B}$ If $f(z)\in \mathcal{T}_{0}(\alpha)$ , then$|z|- \frac{1-2\alpha}{4}|z|^{2}-\frac{1-\sim 9\alpha}{3(3-2\alpha)}|\approx|^{3}\leqq|f(z)|\leqq|z|+\frac{1-2\alpha}{4}|z|^{2}+\frac{1-2\alpha}{3(3-2\alpha)}|z|^{3}$ (3.3)and$1- \frac{1-2\alpha}{2}|z|-\frac{1-\underline{9}\alpha}{3-2\alpha}|z|^{2}\leqq|f’(z)|\leqq 1+\frac{1-2\alpha}{\underline{9}}|z|+\frac{1-2\alpha}{3-2\alpha}|z|^{2}$ (3.4)for $z\in \mathrm{U}$ .Now, we showTheorem 3.1 If $f(z)\in S_{0}(\alpha)$ , then$|z|- \sum_{n=2}^{j}|a_{n}||z|^{n}-A_{j}|z|^{j+1}\leqq|f(z)|\leqq|z|+\sum_{n=2}^{j}|a_{n}||z|^{n}+A_{j}|z|^{j+1}$ (3.5)and1- $\sum_{n=2}^{2j-2}n|a_{n}||z|^{r\iota-1}-B_{j}|\approx|^{2j-2}\leqq|f’(z)|\leqq 1+\sum_{n=2}^{2j-2}.n|a_{n}||z|^{n-1}+B_{j}|z|^{2j-2}$ (3.6)where$A_{l}, \cdot=\frac{1-2\alpha-\sum_{n=2}^{j}\{n-(1+(-1)^{n+1})\alpha\}|a_{n}|}{j+1-(1+(-1)^{j})^{\alpha}}$ $(j\geqq 2)$ (3.7)and$B_{j}=(2j-1) \frac{1-2\alpha-\sum_{n=2}^{2j-2}\{n-(1+(-1)^{n+1})\alpha\}|a_{n}|}{2j-1-2\alpha}$ $(j\geqq 2)$ . (3.8)Proof Note that the coefficient inequalities (1.3) can be written as$\sum_{n=2}^{\infty}\{n-(1+(-1)^{n+1})\alpha\}|a_{r?}|\leqq 1-2\alpha$. (3.9)80This gives us that$\sum_{n=2}^{j}\{n-(1+(-1)^{n+1})\alpha\}|a_{n}|+\{j+1-(1+(-1)^{j})\alpha\}\sum_{n=j+1}^{\infty}|a_{n}|\leqq 1-2\alpha$ (3.10)and$\sum_{n=2}^{2j-2}\{n-(1+(-1)^{n+1})\alpha\}|a_{n}|+(1-\frac{2\alpha}{2j-1})\sum_{n=2j-1}^{\infty}n|a_{n}|\leqq 1-2\alpha$ . (3.11)Therefore, $f(z)\in S_{0}(\alpha)$ satisfies$\sum_{n=j+1}^{\infty}|a_{n}|\leqq A_{j}$ (3.12)and$\sum_{n=2j-1}^{\infty}n|a_{n}|\leqq B_{j}$ . (3.13)Thus, the distortion inequalitity (3.5) follows from (3.12) and the distortion inequality(3.6) follows from (3.13).Remark 3.1 If we take $j=2$ in Theorem 3.1, then we have Theorem A due to Cho,Kwon and Owa [1].Furthermore, we also haveTheorem 3.2 If $f(z)\in \mathcal{T}_{0}(\alpha)_{f}$ then$|z|- \sum_{n=2}^{j}|a_{n}||z|^{n}-C_{j}|z|^{\mathrm{j}+1}\leqq|f(z)|\leqq|z|+\sum_{n=2}^{j}|a_{n}||z|^{n}+C_{j}|z|^{j+1}$ (3.14)and$1- \sum_{n=2}^{j}n|a_{n}||z|^{n-1}-D_{j}|z|^{j}\leqq|f’(z)|\leqq 1+\sum_{n=2}^{j}n|a_{n}||z|^{n-1}+D_{j}|z|^{j}$ (3.15)for $z\in \mathrm{u}$ where$C_{j}= \frac{1-2\alpha-\sum_{n=2}^{j}n\{n-(1+(-1)^{n+1})\alpha\}|a_{n}|}{(j+1)\{j+1-(1+(-1)^{j})\alpha\}}$ $(j\geqq 2)$ (3.16)and$Dj= \frac{1-2\alpha-\sum_{n=2}^{j}n\{n-(1+(-1)^{n+1})\alpha\}|a_{n}|}{j+1-(1+(-1)^{j})\alpha}$ $(j\geqq 2)$ . (3.17)Proof Noting that the coefficient inequalities (1.4) satisfy$\sum_{n=2}^{\infty}n\{n-(1+(-1)^{n+1})\alpha\}|a_{n}.|\leqq 1-2\alpha$ , (3.18)81we have that$\sum_{n=2}^{j}n\{n-(1+(-1)^{n+1})\alpha\}|a_{n}|$$+(j+1) \{j+1-(1+(-1)^{j+2})\alpha\}\sum_{n=j+1}^{\infty}|a_{n}|\leqq 1-2\alpha$ , (3.19)which implies that$\sum_{n=j+1}^{\infty}|a_{n}|\leqq C_{j}$ . (3.20)Further, by virtue of (3.18), we see that$\sum_{n=2}^{j}n\{n-(1+(-1)^{n+1})\alpha\}|a_{n}|+\{j+1-(1+(-1)^{j+2})\alpha\}\sum_{n=j+1}^{\infty}|a_{n}|\leqq 1-2\alpha$ , (3.21)which derives$\sum_{n=j+1}^{\infty}|a_{n}|\leqq D_{j}$ . (3.22)Therefore, the proof of the theorem follows from (3.21) and (3.22).Remark 3.2 If we let $j=2$ in Theorem 3.2, then we have Theorem $\mathrm{B}$ by Cho,Kwon and Owa [1].4 Relation between the classesBy the definitions for the classes $S_{0}(\alpha)$ , and $\mathcal{T}_{0}(\alpha)$ , we know that$S_{0}(\alpha)\subset S_{0}(\beta)\subset S_{0}(0)$ $(0 \leqq\beta\leqq\alpha<\frac{1}{2})$and$\mathcal{T}_{0}(\alpha)\subset \mathcal{T}_{0}(\beta)\subset \mathcal{T}_{0}(0)$ $(0 \leqq\beta\leqq\alpha<\frac{1}{2})$Let us discuss a relation between $S_{0}(\beta)$ and $\mathcal{T}_{0}(\alpha)$ .Theorem 4.1 If $f(z)\in \mathcal{T}_{0}(\alpha)$ , then $f(z) \in S_{0}(\frac{1+2\alpha}{4})$ .Proof Let $f(z)\in \mathcal{T}_{0}(\alpha)$ . Then, if $f(z)$ satisfies$\frac{n-(1+(-1)^{n+1})\beta}{1-2\beta}\leqq n\frac{n-(1+(-1)^{n+1})\alpha}{1-2\alpha}$ (4.1)82for all $n\geqq 2$ , we have that $f(z)\in S_{0}(\beta)$ . which satisfies the inequality (4.1). Aftercalculation (4.1), we have that$\beta\leqq n\frac{n-1+(3+(-1)^{n+1})\alpha}{2n^{2}-(1+(-1)^{n+1})(2n\alpha-2\alpha+1)}$ . (4.2)If $n$ is even, then (4.2) becomes$\beta\leqq\frac{n-1+2\alpha}{2n}$ . (4.3)This implies that$\beta\leqq\frac{1+2\alpha}{4}$ (for even $n$ ) $1$ (4.2)On the other hand, if $n$ is odd, then (4.3) becomes$\beta\leqq\frac{n^{2}-(1-4\alpha)n}{2n^{2}-4n\alpha+4\alpha-2}$ . (4.5)Since, for odd $n$ and $0 \leqq\alpha<\frac{1}{2}$ ,$\frac{n^{2}-(1-4\alpha)n}{2n^{2}-4n\alpha+4\alpha-2}-\frac{1+2\alpha}{4}=\frac{(1-2\alpha)(n-1)(n-1-2\alpha)}{4(n^{2}-2n\alpha+2\alpha-1)}>0$ , (4.6)we conclude that $\beta\leqq\frac{1+2\alpha}{4}$ for all $n$ . Thus we conclude that $\mathcal{T}_{0}(\alpha)\subset S_{0}(\frac{1+^{\underline{\eta}}\alpha}{4})$References[1] N. E. Cho, 0. S. Kwon and S. Owa, Certain subclasses of Sakaguchi functions,SEA Bull Math. 17(1993), 121 –126.[2] K. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Japan 11(1959),72 –75.S. Owa: Department of MathematicsKinki UniversityHigashi-Osaka, Osaka 577-8502Japan$e$-mail: owa@math.h’indai. ac.jpT. Sekine: Office of MathematicsCollege of PharmacyNihon University7-1 Narashinodai, Funabashi-cityChiba, 274-8555, Japan$e$-mail: tsekine@pha.nihon-u. ac.jpR. Yamakawa: Department of MathematicsShibaura Institute of $Te$chnologyMinuma, Saitama-citySaitama $\mathit{3}\mathit{3}7- \mathit{8}\mathit{5}7\mathit{0}_{f}$ Japan$e$ -mail: yamakawa@sic. shibaura-it. ac.jp

Notes on Sakaguchi functions (Coefficient Inequalities in Univalent Function Theory and Related Topics)

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Notes on Sakaguchi functions (Coefficient Inequalities in Univalent Function Theory and Related Topics)

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