Univalence criterion for harmonic mappings and Φ-like functions
Complex Variables and Elliptic Equations
In this paper, we obtain a new characterization for univalent harmonic mappings and obtain a structural formula for the associated function which defines the analytic ф-like functions in the unit disk. The new criterion stated in this article for the injectivity of harmonic mappings implies the well-known results of Kas’yanyuk  and Brickman  for analytic functions, but with a simpler proof than theirs. A number of consequences of the characterization, and examples are also presented. Further investigation provides a new method to construct univalent harmonic mappings with the help of an improved distortion theorem.
Graf, Sergey Yu; Ponnusamy, Saminathan; and Starkov, Victor V., "Univalence criterion for harmonic mappings and Φ-like functions" (2018). Journal Articles. 1103.