Hölder和Minkowski不等式的证明

索引

• 霍尔德不等式
  • 明科夫斯基不等式

Hölder不等式应用分析

Let 1.

Form of Integral
f\in {{L}^{p}}\left[ a,b \right],\text{ }g\in {{L}^{q}}\left[ a,b \right], then:

• f\left( x \right)g\left( x \right)\in {{L}^{1}}\left[ a,b \right]
• \int_{a}^{b}{\left| f\left( x \right)g\left( x \right) \right|dx}\le {{\left( \int_{a}^{b}{{{\left| f\left( x \right) \right|}^{p}}dx} \right)}^{\frac{1}{p}}}{{\left( \int_{a}^{b}{{{\left| g\left( x \right) \right|}^{q}}dx} \right)}^{\frac{1}{q}}}
  or ${{\left| fg \right|}_{1}}\le {{\left| f \right|}