Criteria for univalent functions in the unit disk
Let \(\mathcal A\) denote the class of all normalized analytic functions in the unit disk \(\mathbb D\) and let \(\mathcal S\) be the subset of \(\mathcal A\) consisting of all functions that are univalent in \(\mathbb D\). In the paper the authors study the class \(\mathcal U\) of all functions \(f \in \mathcal A\) such that \(|U_f(z)| < 1\) for \(z \in \mathbb D\), where \[ U_f(z) = \left(\frac{z}{f(z)}\right)^2 f'(z) - 1. \] It is known that each function in \(\mathcal U\) belongs to \(\mathcal S\) and functions in \(\mathcal S\) not necessarily belong to \(\mathcal U\). In the paper two recent results concerning the membership of a function in \(\mathcal U\) are improved. One of them is the following. Let \(g, h \in \mathcal S\) and \(F(z) = g(z)h(z)/z, \,z \in \mathcal U\). Consider the problem of finding the largest value of \(r\) such that the function \(F_r(z) = r^{-1} F(rz)\) belongs to \(\mathcal U\). In a recent paper the authors obtained \(r_0 \approx 0,30294\) and they conjectured that the best possible value of \(r\) is \(1/3\). In the present paper they get \(r \approx 0,3263\) which is closer to the conjectured \(1/3\).
scientific article; zbMATH DE number 6146014
scientific article