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We obtained a version of Hardy-Rellich type inequality in a domain Ω ∈ ℝ  n   which involves the distance to the boundary, the diameter and the volume of Ω. Weight functions in the inequalities depend on the “mean-distance” function and on the distance function to the boundary of Ω. The proved inequalities connect function to first and second order derivatives.

A geometrical version of Hardy-Rellich type inequalities