杨氏不等式:
∀ a , b > 0 forall a,b>0 ∀a,b>0,
a b ≤ a p p + b q q color{red}ab≤frac{a^p}{p}+frac{b^q}{q} ab≤pap+qbq, 其中 1 p + 1 q = 1 frac{1}{p}+frac{1}{q}=1 p1+q1=1.
特别的:
a b ≤ a 2 2 + b 2 2 color{red}ab≤frac{a^2}{2}+frac{b^2}{2} ab≤2a2+2b2
a b ≤ a 2 2 ε + ε b 2 2 color{red}ab≤frac{a^2}{2varepsilon}+frac{varepsilon b^2}{2} ab≤2εa2+2εb2
Cauchy-Schwarz 不等式: