Peetre's inequality

In mathematics, Peetre's inequality, named after Jaak Peetre, says that for any real number ${\displaystyle t}$ and any vectors ${\displaystyle x}$ and ${\displaystyle y}$ in ${\displaystyle \mathbb {R} ^{n},}$ the following inequality holds: $${\displaystyle \left({\frac {1+|x|^{2}}{1+|y|^{2}}}\right)^{t}~\leq ~2^{|t|}(1+|x-y|^{2})^{|t|}.}$$

The inequality was proved by J. Peetre in 1959 and has founds applications in functional analysis and Sobolev spaces.