Logarithmic Holder continuous mappings and Beltrami equation

The paper is devoted to the study of mappings satisfying the inverse Poletsky inequality. We study the local behavior of these mappings, moreover, we are most interested in the case when the corresponding majorant is integrable on some set of spheres of positive linear measure. The most important result is a logarithmic H"{o}lder continuity of such mappings at inner points. As a corollary, we obtained the existence of a continuous $ACL$-solution of the Beltrami equation, which is logarithmic H"{o}lder continuous.