Hardy type inequalities with logarithmic and power weights for a special family of non-convex domains

In the present work we obtain variational Hardy type inequalities with power and logarithmic weights which are generalizations of the corresponding inequalities given earlier in the papers by M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, A. Laptev, and J. Tidblom. We formulate and prove inequalities for arbitrary domains, and then we substantially simplify them for the class of convex domains and a special family of non-convex domains.