摘要
我们知道,柯西不等式:a<sub>i</sub>,b<sub>i</sub>∈R,则sum from i=1 to n a<sub>i</sub><sup>2</sup> sum from i=1 to n b<sub>i</sub><sup>2</sup>≥（sum from i=1 to n a<sub>i</sub>b<sub>i</sub>）<sup>2</sup>……（1）当且仅当a<sub>i</sub>=kb<sub>i</sub>（i=1,2,…,n）不等式等号成立。它可以作如下变形: 由（1）得（sum from i=1 to n a<sub>i</sub><sup>2</sup> sum from i=1 to n b<sub>i</sub><sup>2</sup>）<sup>1/2</sup>≥sum from i=1 to n a<sub>i</sub>b<sub>i</sub>,添项变为sum from i=1 to n a<sub>i</sub><sup>2</sup>+2 （sum from i=1 to n a<sub>i</sub><sup>2</sup> sum from i=1 to n b<sub>i</sub><sup>2</sup>）<sup>1/2</sup>+sum from i=1 to n b<sub>i</sub><sup>2</sup>≥sum from i=1 to n a<sub>i</sub><sup>2</sup>+2sum from i=1 to n a<sub>i</sub>b<sub>i</sub>+sum from i=1 to n b<sub>i</sub><sup>2</sup>,或sum from i=1 to n a<sub>i</sub><sup>2</sup>-2 （sum from i=1 to n a<sub>i</sub><sup>2</sup> sum from i=1 to n b<sub>i</sub><sup>2</sup>）<sup>1/2</sup>+sum from i=1 to n b<sub>i</sub><sup>2</sup>≤sum from i=1 to n a<sub>i</sub><sup>2</sup>-2 sum from i=1 to n a<sub>i</sub> b<sub>i</sub>+sum from i=1 to n b<sub>i</sub><sup>2</sup>,分别配方。
