New criteria for starlikeness in the unit disc

Abstract

It is well-known that the condition ${\operatorname{Re}} \left[1+\frac{zf''(z)}{f'(z)}\right]>0$, $z\in{\mathbb D}$, implies that $f$ is starlike function (i.e. convexity implies starlikeness). If the previous condition is not satisfied for every $z\in {\mathbb D}$, then it is possible to get new criteria for starlikeness by using $\left|\arg\left[\alpha+\frac{zf''(z)}{f'(z)}\right]\right|$, $z\in{\mathbb D}$, where $\alpha>1.$