Journal of the
Korean Mathematical Society JKMS

We obtain the conditions on $\beta$ so that $1+\beta zp'(z)\prec 1+4z/3+2z^{2}/3$ implies $p(z)\prec(2+z)/(2-z)$, $1+(1-\alpha)z$, $(1+(1-2\alpha)z)/(1-z)$, $(0\leq \alpha < 1)$, $\exp(z)$ or $\sqrt{1+z}$. Similar results are obtained by considering the expressions $1+\beta zp'(z)/p(z)$, $1+\beta zp'(z)/p^{2}(z)$ and $p(z)+\beta zp'(z)/p(z)$. These results are applied to obtain sufficient conditions for normalized analytic function $f$ to belong to various subclasses of starlike functions, or to satisfy the condition $|\log(zf'(z)/f(z))|<1$ or $|(zf'(z)/f(z))^{2}-1|<1$ or $zf'(z)/f(z)$ lying in the region bounded by the cardioid $(9 x^2+9 y^2-18x+5)^2- 16 (9 x^2+9 y^2-6x+1)=0$.

Keywords: convex and starlike functions, lemniscate of Bernoulli, subordination, cardioid

MSC numbers: 30C80, 30C45