Nunokawa, Mamoru

Sekine, Tadayuki

Kyoto University Research Information Repository

Title On a sufficient condition for starlikeness (Study on Non-Analytic and Univalent Functions and Applications)Author(s)Nunokawa, Mamoru; Sekine, TadayukiCitation数理解析研究所講究録 (2009), 1626: 83-85Issue Date2009-01URL http://hdl.handle.net/2433/140315RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityOn a sufficient condition for starlikenessMamoru Nunokawa and Tadayuki SekineAbstractWe give a sufflcient condition for a normalized analytic function to bestarlike.Keywoids: $\alpha$-convex functions, starlikle functions, convex functions, univalentfunctions2000 Mathematics Subject Classification. Primaly $30C45$ .1. Introduction and ResultP. T. Mocanu [1] defined $\alpha$-convex functions as follows: Let $f(z)=z+ \sum_{n=2}^{\infty}a_{n}z^{n}$be analytic in the unit disc $\Delta=\{z : |z|<1\}$ , with $f(z)f’(z)/z\neq 0$ there, and let $\alpha$be a real number. Then $f(z)$ is said to be $\alpha$-convex in $\triangle$ if and only if the inequality${\rm Re}[(1- \alpha)\frac{zf^{f}(z)}{f(z)}+\alpha(1+\frac{zf^{\prime l}(z)}{f^{l}(z)})]>0$holds in $\triangle$ . For the functions above, S. S. Miller, P. T. Mocanu and M. O. Read [2]proved the following theorem.Theorm A. If $f(z)=z+ \sum_{n=2}^{\infty}a_{n}z$“ is $\alpha$-convex in the unit disc $\Delta$, then $f(z)$is starlike and univalent in $\triangle$ . Moreover, if $\alpha\geqq 1$ , then $f(z)$ is convex for $l^{z}l<1$ ,and if $\alpha\leqq-1_{f}$ then $1/f(1/z)$ is convex for $|z|>1$ .In this paper, we partly improve Theorem A and to do so, we need the followinglemma [3], [4].Lemma A. Let $p(z)=1+ \sum_{n=1}^{\infty}c_{\eta}z^{n}$ be analytic in the unit disc $\Delta$ and supposedthat there enists a point $z_{0}\in\triangle$ such that${\rm Re}\{p(z)\}>0$ for $|z|<|z_{0}|$ ,${\rm Re}\{p(z_{0})\}=0$ and $p(z_{0})\neq 0$ .Then we have$\frac{z_{0}p’(z_{0})}{p(z_{0})}=ik$1626 2009 83-85 83when$k \geqq\frac{1}{2}(a+\frac{1}{a})$ when $\arg\{p(z_{0})\}=\frac{\pi}{2}$and$k \leqq-\frac{1}{2}(a+\frac{1}{a})$ when $\arg\{p(z_{0})\}=-\frac{\pi}{2}$where $p(z_{0})=\pm ia$ and $a>0$ .$Th\infty rem$. Let $f(z)=z+ \sum_{n=2}^{\infty}a_{n}z^{n}$ be analytic in the unit disc $\triangle$ and let usput$F(z)=[(1- \alpha)\frac{zf^{f}(z)}{f(z)}+\alpha(1+\frac{zf^{\prime l}(z)}{f^{f}(z)})]$ .${\rm Re}\{F(z)\}>0$ in $\Delta$ ,then $f(z)$ is starlik$e$ in $\triangle$ .Proof. For the case $\alpha>0$ , from the hypothesis of the theorem we obtain $f’(z)\neq 0$in $\Delta$ , because if there exists a point $z_{0}\in\triangle$ such that$f’(z_{0})=0$ ,this contradicts the hypothesis of the theorem.Let us put$p(z)= \frac{zf^{f}(z)}{f(z)}$ and $p(z)\neq 0$ in $\Delta$ .Then it follows that$F(z)=p(z)+ \alpha\frac{zp’(z)}{p(z)}$ .If there exists a point $z_{0}\in\Delta$ such that${\rm Re}\{p(z)\}>0$ for $|z|<|z_{0}|$ ,${\rm Re}\{p(z_{0})\}=0$ and $p(z_{0})\neq 0$ ,then from Lemma $A$ , for the case $\arg\{p(z_{0})\}=\pi/2,$ $p(z_{0})=ia$ and $a>0$ , we have$F(z_{0})=p(z_{0})+ \alpha\frac{z_{0}p^{f}(z_{0})}{p(z_{0})}=ia+i\alpha k$and so, $F(z_{0})$ is a pure imaginary value.84Then it follows that${\rm Im}\{F(z_{0})\}=a+\alpha k$$\geqq\frac{1}{2}\{(2+\alpha)c\iota+\frac{\alpha}{a}\}$$\geqq\sqrt{(2+\alpha)\alpha}$.This contradicts the hypothesis of the theorem.For the case $\arg\{p(z_{0})\}=-\pi/2,$ $p(z_{0})=-ia$ and $a>0$ , we have also$F(z_{0})=-ia+i\alpha k$and${\rm Im}\{F(z_{0})\}=-a+\alpha k$$\leqq-a-\frac{1}{2}\alpha(a+\frac{1}{a})$$=- \{(2+\alpha)a+\frac{\alpha}{a}\}$$\leqq-\sqrt{(2+\alpha)(x}$ .This is also contradicts the hypothesis of the theorem and it completes the proof ofthe case $\alpha>0$ .For the case $\alpha<-2$ , applying the same method and Lemma $A$ , we can obtainthe proof of the theorem.Finally, for the case $-2\leqq\alpha<0$ , it depends on [2].References[1] P. T. Mocanu, Une propri\’et\’e de convexit\’e g\’en\’eralis\’ee dans la th\’eorie de larepr\’esentation conforme, Mathematica(Cluj), 11(34), (1969), 127-133, Mr 42#7881.[2] S. S. Miller, P. T. Mocanu and M. O. Read, All $\alpha$-convex functions are univalentand starlike, Proc. Amer. Math. Soc., 37(2), (1973), 553-554.[3] M. Nunokawa, On properties of Non-Carath\’eodory functions, Proc. JapanAcad., 68(6), (1992), 152-153.[4] M. Nunokawa, On the order of strogly starlikeness of strongly convex functions,Proc. Japan Acad., 69(7), (1993), 234-237.798-8, Hoshikuki-machi, Chuo-ku, Chiba-shi, Chiba $2\theta\theta- 0808_{f}$ JapanE-mail:mamoru-nuno@doctor.nifty.jpResearch unit of Mathematics, College of Pharmacy, Nihon University7-1 Namshinodai $7chome_{f}Flmabashi- shi_{1}$ Chiba 274-8555, Japan$E\cdot mail$:sekine.tadayuki@nihon-u.ac.jp85

On a sufficient condition for starlikeness (Study on Non-Analytic and Univalent Functions and Applications)

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On a sufficient condition for starlikeness (Study on Non-Analytic and Univalent Functions and Applications)

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