代数变形工具总结.pdf

o   ?? ?? ?? ? ( ) cyc 1 nn Aa a n= ∑ k ( ) 1 cyc n nn Ga a ⎛⎞=⎜⎟⎝⎠∏ k?( ) ( ) nn Aa G a ≥ ?Z?N? a 1 = a 2 = ! = a n ?H? T? ?? 3 A o? B o? C  ABC △ ? ?k,,xyz ∈R k? 222 2cos 2cos 2cos xyz xy C yz Azx B ++ + + ≥ 3?u? Simon  ?? ? ( ) 1 cyc 1 p p pn n Ma a n ⎛⎞=⎜⎟⎝⎠ ∑ k ( ) ( ) 0 nn Ma Ga = k? ?αβ < ?kMMαβ≤ k ?Z?N? a 1 = a 2 = ! = a n ?H? ? Young  ?? 3 i λ g 1, 2 , , in= L h%? ??Mk?cyc cyc n nn naa λ λ ∑ ∏ ≥ ?Z?N? a 1 = a 2 = ! = a n ?H? ;? | Cauchy  ?? 22 cyc cyc cyc nn n nab a b ⋅ ∑∑∑ ≤ k 2 2 cyc cyc cyc 1 n n nn a a bb ⎛⎞⋅⎜⎟⎝⎠ ∑∑ ∑ ≥ k 2 cyc cyc cyc 1 n n nnn a a bab ⎛⎞⎜⎟⎝⎠ ∑∑ ∑ ≥ k ?Z?N? a 1 b 1 = a 2 b 2 = ! = a n b n ?H? ?6? H ! ! ol de r  ?? 3 1 pq+= k ,0pq > k? 11 cyc cyc cyc pq pq nn n nab a b ⎛⎞⎛⎞ ⋅ ⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠ ∑∑∑ ≤ cyc cyc cyc pq pqnn n nab a b ⎛⎞⎛⎞ ⋅ ⎜⎟⎜⎟⎝⎠⎝⎠ ∑∑∑ ≤ k ?Z?N? a 1 b 1 = a 2 b 2 = ! = a n b n ?H? L~ ?? +L? a 1 a 2 ! a n y b 1 b 2 ! b n k ?L?ycyc nnab ∑ k? ?L?c~?L?y??k"~?L?y?4 ?A?? Chebyshev  ?? +L? a 1 a 2 ! a n k b 1 b 2 ! b n k( ) ( ) nn Aa Ab ⋅ ≤ ( ) nn Aa b ⋅ ?Z?N? a 1 a 2 ! a n = b 1 b 2 ! b n ?H? P ?y ?L? a 1 a 2 ! a n ~k b 1 b 2 ! b n ?~k;" ( ) ( ) n n nn Aa a A bAb ⎛⎞⎜⎟⎝⎠ ≥ f*U??? Minkowski  ?? 3 1 p> k? ( ) ( ) ( ) pn n pn p n Mx y Mx My ++ ≤ 301 p<<k? ( ) ( ) ( ) pn n pn p n Mx y Mx My ++ ≥ ?? Jensen  ?? 3 i λ g i = 1 , 2 , ! , n h%? ??Mk? ?? ( ) fx?k? ( ) cyc cyc nn n n fx fx λλ ⎛⎞⎜⎟⎝⎠∑∑ ≥ ?? ( ) fx?k? ( ) cyc cyc nn n n fx fx λλ ⎛⎞⎜⎟⎝⎠∑∑ ≤ 6 Schur  ?? 3,,xyz + ∈R k 0 r > k? ( )( ) cyc 0 rxx yxz −− ∑ ≥ 333 2 2 2 2 2 2 3 xyz xyzxyyzzxxy yzz