This paper studies a class of generalized harmonic quasiconformal Koebe functions. It is motivated by the shear construction of Clunie and Sheil-Small Ann. Acad. Sci. Fenn. Ser. A I Math. 9: 3--25, 1984 and the harmonic quasiconformal Koebe function. Equivalent univalence conditions, pre-Schwarzian and Schwarzian norms, coefficient inequalities, as well as growth and area theorems for this family of functions are derived. These findings improve several previously known results.
Wang et al. (Sun,) studied this question.