YOKOHAMA MATHEMATICAL

Theorem 1. A function $f(z)=z+\sum_{=2}^{\infty}a_{n}z^{n}$, analytic in $E$, is in the class $S^{\lambda}(\alpha, \beta)$ if and only if (2.3) $f(z)=z$ exp $\{-2\beta(1-\alpha)e^{-\lambda}$ cos $\lambda\int_{0}^{z}\frac{\phi(t)dt}{1+(2\beta-1)t\phi(t)}\}$ , for some $\phi\in A$.

Theorem 1. (iii) $\alpha=0$ and $\beta=(2-coe\lambda)/2$ in Theorem 1 gives the result obtained by Goel [2]. (iv) Replacing $\alpha$ by $(1-\beta+2\alpha\beta)/(1+\beta)$ and $\beta$ by $(1+\beta)/2$ in Theorem 1, we get the corresponding result obtained by Mak\"owka [71. (v) By taking appropriate values of $\alpha,$

Theorem 2. Let $f(z)=z+\sum_{=2}^{\infty}a_{n}z^{n}$ be analytic in E. Then $f(z)\in S^{\lambda}(\alpha, \beta)$, if for some $a\in[0,1$) and $\lambda\in(-\pi/2, \pi/2)$, (3.1) $\sum_{t=2}^{\infty}${$2n(1-\beta)-1+|(1-2\beta)+2\beta(1-\alpha)e^{-\ell\lambda}$ cos $\lambda|$ } $|a_{n}|\leqq 2\beta(1-\alpha)$ cos $\lambda$ , whenever $\beta\in(0,1/2$], (3.2) $\sum_{l=2}^{\infty}${ $(n-1)+|(2\beta-1)(n-1)+2\beta(1-\alpha)e^{-\ell\lambda}$ cos

Corollary 1. A function $f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n}$, analytic in $E$, is a $\lambda$-sPiral-like func- tion of order $a$ if $\sum_{*=2}^{\infty}${$(n-1)+|(n-1)+2(1-\alpha)e^{-\lambda}$ cos $\lambda|$ } $|a_{n}|\leqq 2(1-\alpha)$ cos $\lambda$ , for some $\alpha\in[0,1$), $\lambda\in(-\pi/2, \pi/2)$.

Corollary 2. A function $f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n}$, analytic in $E$, is in the class $H(\lambda)$ if $\sum_{r*=2}^{\infty}$ {$(n-1)+|(1-coe\lambda)(n-1)+(2-coe\lambda)e^{-\lambda}$ cos $\lambda|$ } $|a_{n}|\leqq(2-\cos\lambda)$ cos $\lambda$ , for some $\lambda\in(-\pi/2, \pi/2)$.

Corollary 3. A function $f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n}$, analytic in $E$, is in the class $F_{\lambda,\Pi}$ if for some $\lambda\in(-\pi/2, \pi/2)$, $\sum_{n=2}^{\infty}${$(n-M)+(1-)^{2}+(2M-1)$ cos2 $ \lambda$} $|a_{n}|\leqq(2M-1)$ cos $\lambda$ , whenever $1/2<M\leqq 1$ , $\sum_{n=2}^{\infty}${$(n-1)M+|(M-1)(n-1)+(2M-1)e^{-\ell\lambda}$ cos $\lambda|$ } $|a_{n}|\leqq(2M-1)$ cos $\lambda$ ,

Theorem 3. Let $f(z)=z+\sum_{=2}^{\infty}a_{n}z^{\prime}*$ be analytic in E. If $f(z)\in S^{\lambda}(\alpha, \beta)$, then for $|z|=r_{1}0<r<1$ , and for all $a\in[0,1$), $\beta\in(0,1/2)\cup(1/2,1$], $\lambda\in(-\pi/2, \pi/2)$, (4.1) $|f(z)|\leqq r[\frac{(1+(2\beta-1)r)^{(1\lambda)}-CO8}{(1-(2\beta-1)r)^{(1+\cos\lambda)}}]^{p(1-\alpha)\cos\lambda/(2p-1)}$ , and (4.2) $|f(z)|\geqq r[\frac{(1-(2\beta-1)r)^{(1-\cos\lambda)}}{(1+(2\beta-1)r)^{(1+\cos\lambda)}}]^{\prime(1-\alpha)_{0O8}\lambda/(2\pr$

Corollary 4. If $f(z)=z+\sum_{=2}^{\infty}a_{n}z^{n}$, analytic in $E$, is a $\lambda$-spiral-like function of order $\alpha$, then for $|z|=r<1$ ,

Corollary 5. If $f(z)=z+\sum_{l=2}^{\infty}a_{n}z^{n}$, analytic in $E$, is in $F_{\lambda,K}$, then for $|z|=$ $r\langle 1, M\rangle 1/2,$ $(M\neq 1)$ $\gamma[\frac{(1-((M-1)/M)r)^{(1-\cos\lambda)}}{(1+((M-1)/M)r)^{(1+oos\lambda)}}]^{(2K-1)\cos\lambda/2(K-1)}$ $\leqq|f(z)|\leqq r[\frac{(1+((M-1)/M)r)^{(1-\cos\lambda)}}{(1-((M-1)/M)r)^{(1+\circ\circ\epsilon\lambda)}}]^{(2r-1)0\circ 8\lambda/2(K-1)}$ ; whereas for $M=1$ , $r$ exp (–coe $\lambda\cdot r$) $\leqq|f(z)|\leqq r$ exp (coe $\lambda\cdot$

Theorem 4. Let $f\in S^{\lambda}(\alpha, \beta)$, and $f(z)=z+\sum_{*=2}^{\infty}a_{n}z^{n},$ $z\in E$. (a) If $\beta(1-\alpha)(2-\alpha)$ cos2 $\lambda>(1-\beta)$( $1+(1-\alpha)$ cos2 $\lambda$), let $N=[\frac{\beta(1-\alpha)(2-\alpha)coe^{2}\lambda}{(1-\beta)(1+(1-\alpha)coe^{2}\lambda)}]$ . Then (5.1) $|a_{n}|\leqq\frac{1}{(n-1)!}\prod_{k=2}|(2\beta-1)(k-2)+2\beta(1-\alpha)e^{-\lambda}$ cos $\lambda|$ , for $n=2,3,$ $\cdots,$ $N+2$; and

Theorem 5. $\gamma-s.r$. $S^{\lambda}(\alpha, \beta)$ is the smallest positive root $r$ of the equation (6.3) $(2\beta-1)$($2\beta(1-a)$ cos $(\gamma-\lambda)$ .cos $\lambda-(2\beta-1)$ cos $\gamma$) $r^{2}-2\beta(1-\alpha)$ cos $\lambda\cdot r+coe\gamma=0$ . The result is sharp for the extremal function given in (5.7).

Corollary 6. $\gamma-s.r$. $F_{\lambda.K}$ is the smallest Positive root $r$ of the equation $(M-1)$($(2M-1)$ cos $(\gamma-\lambda)\cdot\cos\lambda-(M-1)$ cos $\gamma$)$r^{2}-(2M-1)M$ cos $\lambda\cdot r+M^{2}$ cos $\gamma=0$ , for $M>1/2$. Taking $a=0$ and $\beta=(2-\cos\lambda)/2$ in Theorem 5, we get the following result.

Corollary 7. $\gamma-s.r$. $H(\lambda)$ is the smallest positive root of the equation (l–cos $\lambda$)( $(2-\cos\lambda)$ cos $(\gamma-\lambda)$ .cos $\lambda-(1-\cos\lambda)$ cos $\gamma$) $r^{2}-(2-coe\lambda)$ cos $\lambda\cdot r+coe\gamma=0$ . When $\gamma=0$ in Corollary 7, we get the radius of starlikeness of $H(\lambda)$ as obtained by Goel [2]. Remarks. Different values of the parameters $\alpha,$ $\beta,$