New criteria for starlikeness in the unit disc

It is well-known that the condition $\mathrm{Re}[1+\frac{z{f}^{\mathrm{\prime }\mathrm{\prime }}(z)}{f^{\mathrm{\prime }}(z)}]>0$, $z\in \mathbb{D}$, 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 $\mid \arg [\alpha +\frac{z{f}^{\mathrm{\prime }\mathrm{\prime }}(z)}{f^{\mathrm{\prime }}(z)}]\mid$, $z\in \mathbb{D}$, where $\alpha >1.$