On the univalence conditions for certain class of analytic functions (Q2445029)

Applying the method of subordination chains [\textit{C. Pommerenke}, Univalent functions. With a chapter on quadratic differentials by Gerd Jensen. Göttingen: Vandenhoeck \& Ruprecht (1975; Zbl 0298.30014)], the authors present an univalence condition for certain class of analytic functions. As one of the main results the following is proved. { Theorem 3.1} Let \(n\) be a positive integer and let \(\lambda\) be a complex number with \(\Re\lambda \leq 0\), \(|\lambda+1/(2n)| > 1/(2n)\). Also, let \(q(z)\) be analytic in the unit disk \(\mathbb{U}\) with \(q(0) =a\), \(q'(0)\neq 0\) and \[ \Re\left(1+\frac{zq''(z)}{q'(z)}\right) > -\frac{1}{n}\Re\left(\frac{1}{\lambda}\right). \] If an analytic function \(p\) is of the form \(p(z)=a+a_nz^n+a_{n+1}z^{n+1}+\cdots\) and satisfies \[ p(z) + \lambda zp'(z) \prec q(z) +\lambda n zq'(z), \] then \(p\prec q\) in \(\mathbb{U}\). From this theorem an univalence condition is deduced: Theorem 3.5. Let \(\lambda\) be a complex number and \(\mu\) a real number such that \(0 < \mu\leq |1+2\lambda|\). If an analytic function \(f\) satisfies an inequality \[ \left|\frac{z^2f'(z)}{(f(z))^2}-\lambda z^2\left(\frac{z}{f(z)}\right)''-1\right| < \frac{\mu}{|1+2\lambda|}, \] then it is univalent in \(\mathbb{U}\).