On regularity theorems for linearly invariant families of harmonic functions

The classical theorem of growth regularity in the class $S$ of analytic and univalent in the unit disc $\Delta$ functions $f$ describes the growth character of different functionals of $f\in S$ and $z\in \Delta$ as $z$ tends to $\partial\Delta.$ Earlier the authors proved the theorems of growth and decrease regularity for harmonic and sense-preserving in $\Delta$ functions which generalized the classical result for the class $S.$ In the presented paper we establish new properties of harmonic sense-preserving functions, connected with the regularity theorems. The effects both common for analytic and harmonic case and specific for harmonic functions are displayed.