Hardy-type inequalities for the Jacobi weight with applications

We prove some new Hardy-type inequalities for the Jacobi weight function. The resulting inequalities contain additional terms with the weight functions characteristic of Poincaré–Friedrichs inequalities. One of the constants in the inequality is unimprovable. We apply the inequalities to extending the available classes of univalent analytic functions in simply-connected domains and find univalence conditions in terms of estimates for the Schwartz derivative of an analytic function on the unit disk, the exterior of the unit disk, and the right half-plane.