Sharp starlikeness conditions for analytic functions with bounded derivative (Q2707062)

The precise range of \(w=f'(z)/f(z)\), \((z\in D)\) is described for the functions \(f\) analytic in the unit disk \(D=\{z: |z|<1\}\), \(f(0)= f'(0)- 1=0\), and \(|f'(z)-1|\leq\lambda\) \((z\in D)\), \(0<\lambda\leq 1\). Several conclusions from that result are obtained. For example, under some restrictions on \(\lambda\), the function \(f\) is strongly starlike \((|\text{arg} w|<\alpha {\pi\over 2}(z\in D)\), \(0<\alpha<1)\) or uniformly starlike \((|w-1|\leq Re w\), \((z\in D))\).