  We prove invariance theorems for general inequalities of different metrics
and apply them to limit relations between the sharp constants in the
multivariate Markov-Bernstein-Nikolskii type inequalities with the polyharmonic
operator for algebraic polynomials on the unit sphere and the unit ball in
$\R^m$ and the corresponding constants for entire functions of spherical type
on $\R^m$. Certain relations in the univariate weighted spaces are discussed as
well.
