Integral inequalities on domains of the Euclidean space for functions with non-zero traces

On plane and space domains, we present several integral inequalities for functions with non-zero 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, Russian Mathematics, 2024, Vol. 68, No. 9, pp. 79–85).
Our theorems are formulated using hyperbolic type domains, the distance from a point to the boundary of a domain and the hyperbolic radius. We give schemes of proofs using the Poincaré metric and its properties, some hyperbolic characteristics of plane domains as well as space domains of hyperbolic type in the sense of Loewner and Nirenberg.