On a new criterion for univalent functions with negative coefficients (Q757637)

The authors show some properties of analytic functions \(f(z)=z- \sum^{\infty}_{k=2}| a_ k| z^ k\) of the unit disk \({\mathbb{D}}\) for which \[ e^{i\alpha}\frac{D^{n+1}f(z)}{z}=(1- \lambda)e^{-i\alpha}+\lambda \cos \alpha (\frac{1+A\omega (z)}{1+B\omega (z)})+i\lambda \sin \alpha, \] where \(-1\leq B<A\leq 1\), \(0<\lambda \leq 1\), \(-\pi /2<\alpha <\pi /2\), \(\omega\) is analytic in \({\mathbb{D}}\) with \(\omega (0)=0\) and \(| \omega (z)| <1\), and \[ D^ nf(z)=\frac{z(z^{n-1}f(z))^{(n)}}{n!} \] (the definition of \(D^ n\) is given incorrectly in the paper).