On a sector for derivative which implies the starlikeness

Abstract It is known that the condition $|\arg f'(z)|<\pi /2$ , $|z|<1$ , is not sufficient for an analytic function $f(z)=z+a_2z^2+\cdots $ in $|z|<1$ , to be starlike with respect to the origin. We look for the largest $\alpha>0$ such that the condition $|\arg f'(z)|<\alpha \pi /2$ in $|z|<1$ is a sufficient condition for f to be a univalent, starlike, convex, or Bazilevic̆ function.