New criteria for starlikeness in the unit disc

TL;DR

The paper introduces new criteria for starlikeness of functions in the unit disc using argument conditions involving the second derivative, extending classical conditions that guarantee starlikeness.

Contribution

It proposes novel argument-based criteria for starlikeness that generalize existing derivative conditions in the unit disc.

Findings

New argument conditions for starlikeness introduced

Criteria applicable when classical conditions are not satisfied

Extension of classical starlikeness criteria

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 $\alpha>1.$