Injectivity conditions for some classes of domains. I, II (Q5947521)

In 1975 F. G. Avhadiev has proved the following general sufficient condition for univalence of the differentiable mapping \(f(z)\) in \( D=\{z:\;\;|z|<1\}\): \[ \begin{aligned} (\star)\qquad &\left|z\varphi_z+\overline{z}\psi_z\right|+\left|z\varphi_{\overline{z}}+ \overline{z}\psi_{\overline{z}}\right|+\left(|f_z-\varphi|+|f_{\overline{z}}-\psi|\right) \frac{|z|^2}{|z|^2-1}\leq\\ &\left(|\varphi|-|\psi|\right)\left(|z|^2-1\right), z\in D,\end{aligned} \] where \(\varphi, \psi\in C^{1}(D)\) and \(|\varphi(z)|>|\psi(z)|\), \(z\in D\) and \(\varphi(\infty)=f_z(\infty)\neq 0\), \(\psi(\infty)=f_{\overline{z}}(\infty)\). A special choice of \(\varphi\) and \(\psi\) gives in particular the well known Becker's condition as well as others known in the topics concerning univalence criteria. In the reviewed paper the author generalized the condition \((\star)\) for some special plane domains \(B\) (\(\infty\in B\)) bounded by a closed Jordan curve. For instance the domains \(G=\mathbb C\setminus\overline{B}\) which are starlike, convex and spirallike are considered. Results of this type seems to be specially interesting in the area of harmonic univalent mappings.