Hardy type inequalities for one weight function and their applications

New one-dimensional Hardy-type inequalities for a weight function of the form $x^\alpha(2-x)^\beta$ for positive and negative values of the parameters $\alpha$ and $\beta$ are put forward. In some cases, the constants in the resulting one-dimensional inequalities are sharp. We use one-dimensional inequalities with additional terms to establish multivariate inequalities with weight functions depending on the mean distance function or the distance function from the boundary of a domain. Spatial inequalities are proved in arbitrary domains, in Davies-regular domains, in domains satisfying the cone condition, in $\lambda$-close to convex domains, and in convex domains. The constant in the additional term in the spatial inequalities depends on the volume or the diameter of the domain. As a consequence of these multivariate inequalities, estimates for the first eigenvalue of the Laplacian under the Dirichlet boundary conditions in various classes of domains are established. We also use one-dimensional inequalities to obtain new classes of meromorphic univalent functions in simply connected domains. Namely, Nehari–Pokornii type sufficient conditions for univalence are obtained.