New criteria for starlikeness in the unit disc

Abstract

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