On a differential test of homeomorphism, found by N.V. Efimov

In the year 1968 N.V. Efimov has proven the following remarkable theorem: \textit{Let $f:{\mathbb{R}}^2\to {\mathbb{R}}^2\in C^1$ be such that $\det f^{\mathrm{\prime }}(x)<0$ for all $x\in {\mathbb{R}}^2$ and let there exist a function $a=a(x)>0$ and constants $C_1⩾0$, $C_2⩾0$ such that the inequalities $\mathrm{\mid }1\mathrm{/}a(x)-1\mathrm{/}a(y)\mathrm{\mid }⩽C_1\mathrm{\mid }x-y\mathrm{\mid }+C_2$ and $|\det f’(x)|\geqslant a(x)|{\rm curl,}f(x)|+a^2(x)$holdtrueforall$x, y\in\mathbb R^2$\mathrm{.}Then$f(\mathbb R^2)$isaconvexdomainand$f$maps$\mathbb R^2$onto$f(\mathbb R^2)$ homeomorhically.} Here $\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{l}f(x)$ stands for the curl of $f$ at $x\in {\mathbb{R}}^2$. This article is an overview of analogues of this theorem, its generalizations and applications in the theory of surfaces, theory of functions, as well as in the study of the Jacobian conjecture and global asymptotic stability of dynamical systems.