Inequalities 모음

Triangle Inequality

$$|A+B|\le |A|+|B|$$

$$|{\int }_0^{t}f(t)g(t)dt|\le {\int }_0^t|f(t)g(t)|dt$$

proof:

$$-|A|\le A\le |A|$$

$$-|B|\le B\le |B|$$

$$-|A|-|B|\le A+B\le |A|+|B|$$

$$|A+B|\le |A|+|B|$$

Cauchy-Schwarz Inequality

$$|⟨\mathbf{\text{v}},\mathbf{\text{w}}⟩{|}^2\le ⟨\mathbf{\text{v}},\mathbf{\text{v}}⟩\cdot ⟨\mathbf{\text{w}},\mathbf{\text{w}}⟩$$

$${|{\int }_0^{t}f(t)g(t)dt|}^2\le |{\int }_0^t{f}^2(t)dt||{\int }_0^t{g}^2(t)dt|$$

Minkowski Ineqautliy

$$||f+g|{|}_p\le ||f|{|}_p+||g|{|}_p$$

$${(\sum _{k=1}^n|x_k+y_k{|}^p)}^{\frac{1}{p}}\le {\left(\sum _{k=1}^n|x_k{|}^p\right)}^{\frac{1}{p}}+{\left(\sum _{k=1}^n|y_k{|}^p\right)}^{\frac{1}{p}}$$

$p=2$일 때

$$\sqrt{\sum _{k=1}^n|x_k+y_k{|}^2}\le \sqrt{\sum _{k=1}^n|x_k{|}^2}+\sqrt{\sum _{k=1}^n|y_k{|}^2}$$

복소수의 triangle inequality를 증명할 때 사용됨.

Jensen's Inequality

Let $\phi$ be a convex function and $X$ be a random variable.

$$\phi (\mathbb{E}(X))\le \mathbb{E}(\phi (X))$$