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.

中文翻译：

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近似理论的夏普常数。二、不同度量的不变性定理和某些多元不等式

我们证明了不同度量的一般不等式的不变性定理，并应用它们来限制多元 Markov-Bernstein-Nikolskii 型不等式中尖锐常数与单位球体上代数多项式和单位球上的多谐算子之间的关系。 m$ 和 $\R^m$ 上球型整个函数的相应常数。还讨论了单变量加权空间中的某些关系。