數學上的切比雪夫總和不等式,或切比雪夫不等式,以切比雪夫命名。
基本介紹
- 中文名:切比雪夫總和不等式
- 外文名:Chebyshev's sum inequality
- 別稱:切比雪夫總額不等式、切比雪夫不等式
- 提出者:切比雪夫
- 套用學科:數學
- 適用領域範圍:數學、不等式
形式,證明,證明一,證明二,積分形式,
形式
它可以比較兩組數積的和及兩組數的線性和的積的大小:
對於兩個實數數列{
}、{
}


若有
,


則有

類似的,若有
,


則有

證明
證明一
考慮和式:

因為有
,
,所以顯然有



將其展開可得

整理可得

反向情況類似,得證。
證明二
因為有
,


所以由排序不等式易知,最大的和為順序和,即:

於是有以下一系列共n個不等式:





將這 n 個不等式分別相加,同時對右式進行因式分解,整理可得:

反向情況可由最小的和為逆序和推得,得證。
積分形式
如果
、
是在[0,1]上的可積實值函式,並且它們同時單增或單減,那么有:



類似的,若


