Integral Inequalities on Domains of the Euclidean Space for Functions with Nonzero Traces

On plane and spatial domains, we present several itnegral inequalities for functions with nonzero boundary trace. These new inequalities for functions are generalizations of the isoperimetric inequalities from the author’s recent paper (F. G. Avkhadiev, “An analog of the Poincaré metric and isoperimetric constants,” Russ. Math. 68 (9), 79–85 (2024). The theorems are formulated using hyperbolic type domains, the distance from a point to the boundary of a domain, and the hyperbolic radius. We give sketches of proofs using the Poincaré metric and its properties, some hyperbolic characteristics of plane domains, as well as spatial domains of hyperbolic type in the sense of Loewner and Nirenberg.