Owa, Shigeyoshi

Sekine, Tadayuki

Yamakawa, Rikuo

Kyoto University Research Information Repository

TitleExamples of Starlike Functions and Convex Functions of orderα (Applications of Complex Function Theory to DifferentialEquations)Author(s)Sekine, Tadayuki; Owa, Shigeyoshi; Yamakawa, RikuoCitation数理解析研究所講究録 (1998), 1062: 120-127Issue Date1998-09URL http://hdl.handle.net/2433/62399RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityExamples of Starlike Functions and Convex Functionsof order $\alpha$Tadayuki SEKINE*[ ]( )Shigeyoshi OWA \dagger [ ] ( )Rikuo YAMAKAWA++[ ] ( )AbstractH.Silverman determines certain coefficient inequalities and distortion theoremsfor univalent functions with negative coefficients that are starlike of order $\alpha$ andconvex of order $\alpha$ . The same coefficient inequalities and the similar distortion theo-rems 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.1 IntroductionLet $A$ denote the class of functions $f(z)$ of the form$f(z)=z+ \sum_{=n2}a_{k}Z^{k}\infty$ $(n\in \mathrm{N}=\{1,2,3, \cdots\})$ (1)that are analytic in the unit disk $U=\{z : |z|<1\}$ .We first consider the so-called subclasses of analytic functions with negative coeffi-cients. Let $A(n)$ denote the subclass of $A$ consisting of functions of the form$f(z)=z- \sum_{+k=n1}^{\infty}a_{k}z^{k}$ $(a_{k}\geqq 0, n\in \mathrm{N}=\{1,2,3, \cdots\})$ . (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$\mathrm{t}$ Department of Mathematics, Shibaura Institute of Technology, 307, Shinsaku, Omiya-shi 330, Japan1062 1998 120-127 120and such a subclass of $A(n)$ consisting of all the starlike functions of order $\alpha$ is denotedby $T_{\alpha}(n)$ . Also, $f(z)\in T(n)$ is said to be convex of order $\alpha(0\leqq\alpha<1)$ if and only if itsatisfies${\rm Re} \{1+\frac{zf^{J/}(_{\mathcal{Z}})}{f’(z)}\}>\alpha$ $(z\in U)$ (4)and the subclass by $C_{\alpha}(n)$ . The classes $T(n),$ $T_{\alpha}(n)$ and $C_{\alpha}(n)$ were introduced by$\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{j}\mathrm{e}\mathrm{a}[1]$ and these classes have been studied by Srivastava, Owa and $\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{j}\mathrm{e}\mathrm{a}[9]$ ,Kiryakova, Saigo and Owa [2], and $\mathrm{S}\mathrm{e}\mathrm{k}\mathrm{i}\mathrm{n}\mathrm{e}[4]$ .For $n=1$ , these notations are usually used as $T_{\alpha}(1)=T^{*}(\alpha),$ $C_{\alpha}(1)=C(\alpha)$ whichwere introduced by $\mathrm{S}\mathrm{i}\mathrm{l}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{n}[8]$ .Using the same way of $\mathrm{S}\mathrm{i}\mathrm{l}\mathrm{V}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{a}\mathrm{n}[8],$ $\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{j}\mathrm{e}\mathrm{a}[1]$ determined a necessary and suffi-cients conditions for a function in $A(n)$ belongs to $T_{\alpha}(n)$ and $C_{\alpha}(n)$ .Theorem $\mathrm{A}(\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{j}\mathrm{e}\mathrm{a}[1])$ . A function $f(z)$ in $A(n)$ is in $T_{\alpha}(n)$ if and only if$\sum_{k=n+1}^{\infty}(k-\alpha)a_{k}\leqq 1-\alpha$ $(0\leqq\alpha<1)$ . (5)Theorem $\mathrm{B}(\mathrm{C}\mathrm{h}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{j}\mathrm{e}\mathrm{a}[1]).$ A function $f(z)$ in $A(n)$ is in $C_{\alpha}(n)$ if and only if$\sum_{k=n+1}^{\infty}k(k-\alpha)a_{k}\leqq 1-\alpha$ $(0\leqq\alpha<1)$ . (6)Recently, in [5] we introduced the subclass $A(n, \theta)$ of $A$ , and the subclasses $T_{\alpha}^{*}(n, \theta)$and $C_{\alpha}(n, \theta)$ of $A(n, \theta)$ in the following manner.Let $A(n, \theta)$ denote the subclass of $A$ consisting of functions of the form$f(z)=z- \sum_{k=n+1}^{\infty}e-1\theta a_{k^{Z}}i(k)k$ $(a_{k}\geqq 0, n\in \mathrm{N})$ . (7)We note that $A(n, 0)=A(n)$ , that is, $A(n, 0)$ is the subclass of analytic functionswith negative coefficients. We denote by $T_{\alpha}^{*}(n, \theta)$ and $C_{\alpha}(n, \theta)$ the subclasses of $A(n, \theta)$of starlike and convex functions of order $\alpha$ in $U$ , respectively.$\mathrm{t}\prime \mathrm{V}\mathrm{e}$ proved the same coefficients inequalities as Chatterjea showed for $f(z)$ in $A(n, \theta)$as follows.Theorem $\mathrm{C}$ (Sekine and Owa [5]). A function $f(z)$ in $A(n, \theta)$ is in $T_{\alpha}^{*}(n, \theta)$ if andonly if$k=n+ \sum_{1}^{\infty}(k-\alpha)a_{k}\leqq 1-\alpha$ $(0\leqq\alpha<1)$ . (8)Theorem $\mathrm{D}$ (Sekine and Owa [5]). A function $f(z)$ in $A(n, \theta)$ is in $C_{\alpha}(n, \theta)$ if andonly if$k=n+ \sum_{1}^{\infty}k(k-\alpha)a_{k}\leqq 1-\alpha$ $(0\leqq\alpha<1)$ . (9)121Further we determined the following Distortion theorems.Theorem $\mathrm{E}$(Sekine and Owa [5]). If $f(z)$ is in $T_{\alpha}^{*}(n, \theta)$ , then$|z|- \frac{1-\alpha}{n+1-\alpha}|z|^{n+1}\leq|f(z)|\leq|z|+\frac{1-\alpha}{n+1-\alpha}|z|^{n+1}$ . (10)The right-hand equality holds for the function$in\theta$ 1 $-\alpha$$f(z)=z-e$ $\overline{n+1-\alpha}z^{n+1}$ $(z=re^{-i(\theta+} \frac{\pi}{n}),$ $r<1)$ (11)and the left-hand equality holds for the function$in\theta$$1-\alpha$$f(z)=z-e$ $\overline{n+1-\alpha}^{\mathcal{Z}^{n+1}}$ $(z=re^{-i\theta}, r<1)$ . (12)Theorem $\mathrm{F}$ (Sekine and Owa [5]). If $f(z)$ is in $C_{\alpha}(n, \theta)$ , then$|z|- \frac{1-\alpha}{(n+1)(n+1-\alpha)}|z|^{n+1}\leq|f(z)|\leq|z|+\frac{1-\alpha}{(n+1)(n+1-\alpha)}|z|^{n+1}$ . (13)The right-hand equality holds for the function$f(z)=z-e^{in\theta_{\frac{1-\alpha}{(n+1)(n+1-\alpha)}z^{n+1}}}$ $(z=re^{-i(} \theta+\frac{\pi}{n}),$ $r<1)$ (14)and the left-hand equality holds for the function$f(z)=z-e^{in\theta_{\frac{1-\alpha}{(n+1)(n+1-\alpha)}}1}z^{n+}$ $(z=re^{-i\theta}, r<1)$ . (15)2 ExamplesLet $A_{\alpha}(n, \theta, h)$ denote the subclass of $A(n, \theta)$ consisting of functions of the form$f(z)=z- \sum_{k=n+1}^{\infty}ei(k-1)\theta kak,h^{\mathcal{Z}}$ $(h\geqq-n)$ , (16)$(1-\alpha)^{2}$where$a_{k,h}=\overline{(k+h-\alpha)(k+1+h-\alpha)(k-\alpha)}$$(0\leqq\alpha<1)$ .Let $B_{\alpha}(n, \theta, h)$ denote the subclass of $A(n, \theta)$ consisting of functions of the form$g(z)=z- \sum_{=kn+1}^{\infty}e^{i}b(k-1)\theta k,hz^{k}$ $(h\geqq-n)$ , (17)where $b_{k,h}= \frac{(1-\alpha)^{2}}{(k+h-\alpha)(k+1+h-\alpha)(k-\alpha)k}$ $(0\leqq\alpha<1)$ .Then we have the following theorems.122Theorem 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$ , $h=-n$ ,$\frac{(1-\alpha)^{2}}{n+1+h-\alpha}<\frac{(1-\alpha)^{2}}{1-\alpha}--1-\alpha$ , $h>-n$ .Hence we know that $f(z)$ is an element of $T_{\alpha}^{*}(n, \theta)$ by virtue of the theorem C. $\square$Using Theorem $\mathrm{B}$ , we can also prove the following theorem 2.2.Theorem 2.2. If $g(z)\in B_{\alpha}(n, \theta, h)$ , then $g(z)\in C_{\alpha}(n, \theta)$ .In the case of $\theta=0$ , these theorems were proved by Sekine and Yamanaka [6].Example 2.1. If $f(z) \in A_{0}(1, \frac{\pi}{4}, -1)_{2}$ then we have$f(z)$ $=$ $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 withMathematica. In view of the figures, we can image that the functions of the examples2.1 and 2.2 are starlike function and convex function, respectively.123Figure 1: Image of $|z|\leqq 1$ by $f(z)=z- \sum_{k2}^{9}=\frac{1}{(k-1)k^{2}}e^{i}(k-1)\frac{\pi}{4}xk$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$124Further we estimate the functions in $T_{\alpha}^{*}(n, \theta)$ and $C_{\alpha}(n, \theta)$ . Let $f(z) \in T_{0}^{*}(1, \frac{\pi}{4})$ , thenby the theorem $\mathrm{E}$ we have$|z|- \frac{1}{2}|z|^{2}\leqq|f(z)|\leqq|z|+\frac{1}{2}|Z|^{2}$ . (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$z=re^{-(\pi+\frac{\pi}{4})}$ , also the left-hand equality on $z=re^{-i(\frac{\pi}{4})}$ . By letting $rarrow 1$ for thefunction $f(z)=z- \frac{1+i}{2\sqrt{2}}z^{2}$ , we have $\frac{1}{2}\leqq|f(Z)|\leqq\frac{3}{2}$ .Figure 3: $\mathrm{I}\mathrm{m}\mathrm{a}_{\iota\supset}\sigma \mathrm{e}$ of $|z|\leqq 1$ by $f(Z)=z- \frac{1+i}{2\sqrt{2}}z^{2}$Figure 4: Images of $|z|\leqq 1$ by $z- \frac{1+i}{2^{\sqrt{2}}}z^{2},$ $\frac{z}{2}$ and $\frac{3}{2}z$125Also, in case of $g(z) \in C_{0}^{*}(1, \frac{\pi}{4})$ , we have $\frac{3}{4}\leqq|g(z)|\leqq\frac{5}{4}$ .Figure 5: image of $|z|\leqq 1$ by $g(z)=z- \frac{1+i}{4\sqrt{2}}z^{2}$Figure 6: Images of $|z|\leqq 1$ by $z- \frac{1+i}{4\sqrt{2}}z^{2},$ $\frac{3}{4}z$ 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), 25-46.126[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 $\alpha$ withnegative coefficients, Math. Sci. Res. Hot-Line 1(1997), 7-12.[7] A. Schild and H. Silverman, Convolutions of univalent functions with negative coeffi-cients, 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 starlikefunctions, Rend. Sem. Mat. Univ. Padova 77(1987), 115-124.127

Examples of Starlike Functions and Convex Functions of order α (Applications of Complex Function Theory to Differential Equations)

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Examples of Starlike Functions and Convex Functions of order α (Applications of Complex Function Theory to Differential Equations)

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