绝对值三角不等式
[large ||a|-|b||leq|apm b|leq |a|+|b|
]
均值不等式
[large dfrac{1}{dfrac1n (dfrac1{a_1}+dfrac1{a_2}+dots dfrac1{a_n})}leq sqrt[n]{a_1a_2dots a_n}leq dfrac{a_1+a_2+dots+a_n}{n}
]
当且仅当 (a_1=a_2=dots =a_n) 时等号成立
已知 (a_i> 0)
伯努利不等式
[large (1+x)^nge 1+nx quad(forall x>-1 ,nin mathbb{N}^*)
]
对 (xge 0,yge 0,nin mathbb{N}^*) ，有
[large (x+y)^n ge x^n+y^n,qquad (x^n+y^n)^{frac 1n}le x+y
]
[large (x+y)^{frac 1n}le x^{frac 1n}+y^{frac 1n},qquad |x^{frac 1n}-y^{frac 1n}|le |x-y|^{frac 1n}
]
因式分解
[large a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+dots+ab^{n-2}+b^{n-1})
]
闵可夫斯基不等式
[large left(sumlimits_{i=1}^n(a_i+b_i)^2
ight)^{1/2}le left(sumlimits_{i=1}^na_i^2
ight)^{1/2}+left(sumlimits_{i=1}^nb_i^2
ight)^{1/2}
]