부산대학교 도서관

Let $S^\ast(p)$ denote the class of all $p$-valently starlike analytic functions $f$ in the unit disc $U=\{z\colon |z|<1\}$ of the form $f(z)=z^p+\sum^\infty_{n=p+1} a_n z^n$. When $p=1$, $S^\ast(1)=S^\ast$, the class of starlike univalent analytic functions on $U$. If $f(z)$ of the above form satisfies $|(zf''(z)/f'(z))+1-p|< (\sqrt{2}/8)(5p+4\sqrt{p}+4)$, $z\in U$, it is proved that $f\in S^\ast (p)$. For $p=1$, the condition reduces to $|zf''(z)/f'(z)|< 13\sqrt{2}/8=2.298\cdots$, for $f\in S^\ast$. This is an improvement of earlier results due to \n R. Singh\en and \n S. Singh\en, and \n S. S. Miller\en and \n P. T. Mocanu\en.