# 数学分析笔记——积分不等式

## 数学分析笔记——积分不等式

Contents

### 1​ Cauchy-Schwarz 不等式

$$\left( \int_{a}^{b}{f \left( x \right)g \left( x \right)}\mathrm{d}x \right)^2\leq\int_{a}^{b}{f^2 \left( x \right)}\mathrm{d}x\int_{a}^{b}{g^2 \left( x \right)}\mathrm{d}x.$$

$$\int_{a}^{b}{ \left( tf \left( x \right)-g \left( x \right) \right)^2}\mathrm{d}x\geq0,$$

$$\left( \int_{a}^{b}{f^2 \left( x \right)}\mathrm{d}x \right)t^2-2 \left( \int_{a}^{b}{f \left( x \right)g \left( x \right)}\mathrm{d}x \right)t+\int_{a}^{b}{g^2 \left( x \right)}\mathrm{d}x\geq0$$

$$4 \left( \int_{a}^{b}{f \left( x \right)g \left( x \right)}\mathrm{d}x \right)^2-4\int_{a}^{b}{f^2 \left( x \right)}\mathrm{d}x\int_{a}^{b}{g^2 \left( x \right)}\mathrm{d}x\leq0,$$

### 2​ Young 不等式

$$ab\leq\int_{0}^{a}{f \left( x \right)}\mathrm{d}x+\int_{0}^{b}{f^{-1} \left( x \right)}\mathrm{d}x.$$

$2'$ 推论: 若 $u,v\geq0,p,q>0,\frac{1}{p}+\frac{1}{q}=1$, 则
$$uv\leq\frac{u^p}{p}+\frac{v^q}{q}.$$

### 3​ Hölder不等式

$$\int_{a}^{b}{|f \left( x \right)g \left( x \right)|}\mathrm{d}x\leq \left( \int_{a}^{b}{|f \left( x \right)|^p}\mathrm{d}x \right)^{\frac{1}{p}} \left( \int_{a}^{b}{|g \left( x \right)|^q}\mathrm{d}x \right)^{\frac{1}{q}}.$$

$$\int_{a}^{b}{\left|\frac{f \left( x \right)}{ \left( \int_{a}^{b}{|f \left( x \right)|^p}\mathrm{d}x \right)^{\frac{1}{p}}}\frac{g \left( x \right)}{ \left( \int_{a}^{b}{|g \left( x \right)|^q}\mathrm{d}x \right)^{\frac{1}{q}}}\right|}\mathrm{d}x\leq\int_{a}^{b}{ \left( \frac{|f \left( x \right)|^p}{p\int_{a}^{b}{|f \left( x \right)|^p}\mathrm{d}x}+\frac{|g \left( x \right)|^q}{q\int_{a}^{b}{|g \left( x \right)|^q}\mathrm{d}x} \right)}\mathrm{d}x=\frac{1}{p}+\frac{1}{q}=1 ,$$

### 4 Minkowski 不等式 $\left( p>1 \right)$

$$\left( \int_{a}^{b}{|f \left( x \right)+g \left( x \right)|^p}\mathrm{d}x \right)^{\frac{1}{p}}\leq \left( \int_{a}^{b}{|f \left( x \right)|^p}\mathrm{d}x \right)^{\frac{1}{p}}+ \left( \int_{a}^{b}{|g \left( x \right)|^p}\mathrm{d}x \right)^{\frac{1}{p}}.$$

$$\begin{split} \left( \int_{a}^{b}{|f \left( x \right)+g \left( x \right)|^p}\mathrm{d}x \right)^{\frac{1}{p}\cdot p}&=\int_{a}^{b}{|f \left( x \right)+g \left( x \right)||f \left( x \right)+g \left( x \right)|^{p-1}}\mathrm{d}x\\ &\leq\int_{a}^{b}{|f \left( x \right)||f \left( x \right)+g \left( x \right)|^{p-1}}\mathrm{d}x+\int_{a}^{b}{|g \left( x \right)||f \left( x \right)+g \left( x \right)|^{p-1}}\mathrm{d}x\\ &\leq \left( \int_{a}^{b}{|f \left( x \right)|^p}\mathrm{d}x \right)^{\frac{1}{p}} \left( \int_{a}^{b}{|f \left( x \right)+g \left( x \right)|^{ \left( p-1 \right)q}}\mathrm{d}x \right)^{\frac{1}{q}}\\ &\quad+ \left( \int_{a}^{b}{|g \left( x \right)|^p}\mathrm{d}x \right)^{\frac{1}{p}} \left( \int_{a}^{b}{|f \left( x \right)+g \left( x \right)|^{ \left( p-1 \right)q}}\mathrm{d}x \right)^{\frac{1}{q}}\\ &= \left( \left( \int_{a}^{b}{|f \left( x \right)|^p}\mathrm{d}x \right)^{\frac{1}{p}}+ \left( \int_{a}^{b}{|g \left( x \right)|^p}\mathrm{d}x \right)^{\frac{1}{p}} \right) \left( \int_{a}^{b}{|f \left( x \right)+g \left( x \right)|^{p}}\mathrm{d}x \right)^{\frac{1}{q}}.\\ \end{split}$$

### 6​ Chebyshev 不等式

$$\int_{a}^{b}{f \left( x \right)g \left( x \right)}\mathrm{d}x\geq\frac{1}{b-a}\int_{a}^{b}{f \left( x \right)}\mathrm{d}x\int_{a}^{b}{g \left( x \right)}\mathrm{d}x.$$