Yamakawa, Rikuo

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Title On some conditions of starlikenessAuthor(s)Yamakawa, RikuoCitation数理解析研究所講究録 (1999), 1112: 138-141Issue Date1999-09URL http://hdl.handle.net/2433/63350RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityOn some conditions of starlikenessRikuo YamakawaDepartment of natural SciencesShibura Institute of technologyAbstractLet $f(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots$ be analytic in the unit disk $U$ .Yuan Chun Fang proved that$| \frac{f^{\iota/}(z)}{f’(z)}|<m\Rightarrow Re\frac{zf’(Z)}{f(z)}>0$ $(z\in U)$ ,where $m(=2.83\cdots)$ is the best possible. In this paper, we generalizethis theorem.1. IntroductionLet $A_{p}$ denote the class of functions of the form(1) $f(z)=z^{p}+ap+1^{Z+az^{p2}}p+1\mathrm{P}+2+\ldots\ldots$ $(p\in N)$ ,which are analytic in the unit disk $U=\{z : |z|<1\}$ , and $M_{p}$ denote theclass of functions(2) $f(z)=z-p+a-p+1^{Z+a}-p+2z^{-_{\mathrm{P}}}-p+1+2\ldots\ldots$ $(p\in N)$ ,which are analytic in punctured disk $U-\{0\}$ . Then we denote two classesof starlike functions as follows:(3) $A_{p}^{*}= \{f\in Ap : \Re\frac{zf^{\mathfrak{l}}(z)}{f(z)}>0, z\in U\}$ ,(4) $M_{\mathrm{p}}^{*}= \{f\in M\mathrm{p} : \Re\frac{zf’(Z)}{f(z)}<0, z\in U\}$ .For the class $A_{1}$ , Singh and Singh [4] showed the following theorem.1112 1999 138-141 138Theorem A Let $f(z)\in A_{1}$ , then(5) $| \frac{f’’(z)}{f’(z)}|<\frac{3}{2}$ $(z\in U)\Rightarrow f(z)\in A_{1}^{*}$ .He used Jack’s Lemma. Mocanu $[3,\mathrm{p}.338]$ showedTheorem $\mathrm{B}$ Let $g(z)= \frac{\mathrm{e}^{\lambda z}-1}{\lambda}$ .(6) $g(z)\in A_{1}^{*}\Leftrightarrow|\lambda|\leq m=2.8329\cdots$ .Where $m$ is the least positive solutin of the following equation(7) $\cos\sqrt{x^{2}-1}+\sqrt{x^{2}-1}\sin\sqrt{x^{2}-1}-\frac{1}{e}=0$ .Miller and Mocanu [2] proved, using their theory of first order differentialsubordination, thatTheorem $\mathrm{C}$ Let $f(z)\in A_{1}$ . Then(8) $| \frac{f’’(z)}{f’(z)}|<2\Rightarrow f(z)\in A_{1}^{*}$ .In the same article [2], they posed the interesting question of finding themaximum value of $k$ for which(9) $| \frac{f’’(z)}{f’(z)}|<k\Rightarrow f(z)\in A_{1}^{*}$ .From the above two theorems, $2\leq k\leq m$ . Some Mathematicians im-proved the lower bound of $k$ . And recently, Yuan Chun Fang [4] showedTheorem $\mathrm{D}$ Let $f(z)\in A_{1}$ . Then(10) $| \frac{f’’(z)}{f^{l}(z)}|<m$ $(z\in U)\Rightarrow f(z)\in A_{1}^{*}$ .The result is sharp, with the extremal function(11) $G(z)= \frac{e^{mz}-1}{m}$ .The purpose of this paper is to obtain simillar theorems for $A_{p}$ and $M_{p}$ .139Theorem 1 If $f(z)\in A_{p}$ satisfies(12) $| \frac{f^{\prime/}(z)}{f’(z)}-\frac{p-1}{p}\frac{f’(z)}{f(z)}|\leq m$ $(z\in U)$ ,then $f$ belongs to $A_{p}^{*}$ . The result is $sh\mathrm{a}rp$, with the extremal function(13) $G_{1}(z)=( \frac{e^{mz}-1}{m})^{p}$ .Theorem 2 If $f(z)\in M_{\mathrm{p}}$ satisfies(14) $| \frac{f^{\prime/}(z)}{f’(z)}-\frac{p+1}{p}\frac{f’(z)}{f(z)}|\leq m$ $(z\in U)$ ,then $f$ belongs to $M_{p}^{*}$ . The result is sharp, with the extremal $fu$nction(15) $c_{2}(z)=( \frac{m}{e^{mz}-1})\mathrm{P}$ .2. Proof of Theorem 1We use the following lemma due to Miller and Mocanu [2, Theorem 2].Lemma let $h$ be $con$vex in $U$ and $\theta$ and $\phi$ be analytic in a domain $D$ .Let $p$ be $\mathrm{a}nal_{\mathrm{J}’}tiC$ in $U$ , with $p(\mathrm{O})=h(\mathrm{O})=\theta(p(\mathrm{O}))$ and $p(U)\subset D.$ If th$\mathrm{e}$differential equation(16) $\theta(q(\mathcal{Z}))+zq(\prime z)\phi(q(Z))=h(z)$$h$as a $uni$valent solution in $U$ that satisfies $q(\mathrm{O})=h(0)$ an$d$(17) $\theta(q(z))\prec h(z)$ ,then the $\mathrm{r}el$ation(18) $\theta(_{P}(Z))+Zp’(Z)\phi(p(Z))\prec h(z)$implies $p(z)\prec q(z)$ . The function $q$ is the best dominan$\mathrm{t}$ of (18).Suppose $f(z)\in A_{p^{\mathrm{S}\mathrm{a}\mathrm{t}}}^{*}\mathrm{i}\mathrm{S}\mathrm{f}\mathrm{i}\mathrm{e}\mathrm{S}(12)$, then we have(19) $\frac{zf’’(z)}{f(z)},-\frac{p-1}{p}\frac{zf’(Z)}{f(z)}\prec mz$ .140Let put$p(Z)= \frac{1}{p}\frac{zf’(Z)}{f(z)}$ , $q(Z)= \frac{1}{p}\frac{zG_{1}’(z)}{G_{1}(z)}$$h(z)=1+mz$ , $\theta(z)=z$ , and $\phi(z)=\frac{1}{z}$ .Then we have(20) $q(z)= \frac{zG’(Z)}{G(z)}=m\frac{ze^{mz}}{e^{mz}-1}$ .(21) $q(_{Z})+ \frac{zq’(z)}{q(z)}=1+mz$ ,and(22) $p(z)+ \frac{zp’(z)}{p(z)}=1+\frac{zf’’(Z)}{f(z)},-\frac{p-1}{p}\frac{zf’(Z)}{f(z)}$.From (19) and (22) we obtain (18), and from (21) we obtain (16). It yields$p(z)\prec q(z)$ , and so $\frac{zf’(z)}{f(z)}\prec\frac{zG^{l}(z)}{G(z)}$ . Therefore, from Theorem $\mathrm{B}$ we obtainthat$\frac{zf’(Z)}{f(z)}>0$ $(z\in U)$ .Concernig the exremal function, we have$\frac{zG_{1}^{\mathrm{f}\prime}(z)}{G_{1}(_{Z)}},-\frac{p-1}{p}\frac{G_{1}f’(Z)}{G_{1}(_{Z)}}=m$ .This implies that $G_{1}(z)$ is exremal.Proof of Theorem 2 is similar, so we omit.References[1] Yuan Chun Fang, Univalence of analytic solutions to some first orderdifferential equations, Acta Mathematica Sinica (Chinese, Chinese sum-mary). 35(4), 483-491.[2] S.S. Miller and P.T. Mocanu, On some class of first-order differentialsubordinations, Michigan Math. J. 32(1985), 185-194.[3] P.T. Mocanu, Asupra razei de stelaritate a functiilor univalente, Stud.Cer. Mat. (Cluj), 11(1960), 337-341.[4] R.Singh and S. Singh, Starlikeness and convexity of certain integrals,Ann. Univ. Ma$7\dot{\eta}ae$ Curie-Sklodowska., 35(1981), 145-147. Colloq. Math.47(1982), 309-314.141

On some conditions of starlikeness

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On some conditions of starlikeness

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