Properties of the mappings that are close to the harmonic mappings. II

We continue studying the mappings that are close to the harmonic mappings ($\varepsilon$-quasiharmonic mappings with $\varepsilon$ small). This study originates with the previous articles of the author. The results of the article include a theorem on connection between the notion of $\varepsilon$-quasiharmonic mapping and the solutions to Beltrami systems, an analog to the arithmetic mean property of harmonic functions for $\varepsilon$-quasiharmonic mappings, a theorem on stability in the Poisson formula for harmonic mappings in the ball, and a theorem on the local smoothing of $\varepsilon$-quasiharmonic mappings with $\varepsilon$ small which preserves proximity to the harmonic mappings.