Estimating the argument of some analytic functions (Q1352926)

Let \(f\) be analytic in \(\mathbb{D}\), with \(f(0)=f'(0)-1=0\), and let \(\partial f(\mathbb{D})\) be contained in a ring \(\{w: m_f\leq |w|\leq M_f\}\). Putting \(\rho_f=M_f/m_f\), the authors are interested in estimating \(|\arg(f(z)/z)|\) in terms of \(\rho_f\) plus certain assumptions on \(\text{Re }{zf'(z)\over f(z)}\). Improvements over earlier results [the authors, Complex Variables, Theory Appl. 26, No. 3, 191-202 (1994; Zbl 0839.30006)] are obtained.