The radius of univalence for certain class of analytic functions (Q2751162)

Let \(f(z)=z+a_2z^2+\cdots\) be an analytic function satisfying \(\text{ Re}(f(z)/z)^\alpha > \sigma\), \(|z|<1\), \(\alpha>0\), \(\sigma<1\) and \(p(z)=1+p_1z+\cdots\) be a function with positive real part in \(|z|<1\). Lower bounds for \(\text{ Re}(p(z)+czp'(z))\), \(c>0\), and \(\text{ Re}[(zf'(z)/f(z))(f(z)/z)^\alpha]\) are computed. Using this, the radius of univalence of \(f(z)\) is computed. The authors have also proved the distortion theorem for \(f(z)\).NEWLINENEWLINEFor the entire collection see [Zbl 0966.00026].