Bulletin of the
Korean Mathematical Society BKMS

We consider a sufficient condition for $w(z)$, analytic in $|z|<1$, to be bounded in $|z|<1$, where $w(0)=w'(0)=0$. We apply it to the meromorphic starlike functions. Also, a certain Briot-Bouquet differential subordination is considered. Moreover, we prove that if $p(z)+zp'(z)\phi(p(z))\prec h(z)$, then $p(z)\prec h(z)$, where $h(z)=[(1+z)(1-z)]^{\alpha}$, under some additional assumptions on $\phi(z)$.

Keywords: analytic, meromorphic, convex, starlike, univalent, Nunokawa's lemma, Briot-Bouquet, differential subordination

MSC numbers: Primary 30C45; Secondary 30C80