Linear combinations of univalent harmonic mappings with complex coefficients

D. Khurana, R. Kumar, S. Gupta, S. Singh

We study the linear combinations $f(z)= \lambda f_{1}(z)+(1-\lambda) f_{2}(z)$ of two univalent harmonic mappings $f_{1}$ and $f_{2}$ in the cases when $\lambda$ is some complex number. We determine the radius of close-to-convexity of $f$ and establish some sufficient conditions for $f$ to be locally-univalent and sense-preserving. Some known results reduce to particular cases of our general results.