一道乘积不等式与解答

2023-03-17 13:38 作者:Ethan骁 0人读过 | 我要投稿

给定 $n%5Cgeq%202$ , 对于任意 $a_1%2Ca_2%2C%5Ccdots%20%2Ca_n%5Cin%20(-1%2C1)$ , 求证:

$%5Cprod%5Climits_%7Bi%3D1%7D%5En%5Cprod%5Climits_%7Bj%3D1%7D%5En%5Cfrac%7B1%2Ba_ia_j%7D%7B1-a_ia_j%7D%5Cgeqslant%201$ . (韦东奕供题)

注意到

$%5Cbegin%7Baligned%7D%0A%5Cln%5Cprod%5Climits_%7Bi%3D1%7D%5En%5Cprod%5Climits_%7Bj%3D1%7D%5En%5Cfrac%7B1%2Ba_ia_j%7D%7B1-a_ia_j%7D%0A%26%3D%5Csum%5Climits_%7Bi%3D1%7D%5En%5Csum%5Climits_%7Bj%3D1%7D%5En%5Cln%5Cleft(%201%2Ba_ia_j%5Cright)%20-%5Cln%5Cleft(%201-a_ia_j%5Cright)%5C%5C%0A%26%3D%5Csum%5Climits_%7Bi%3D1%7D%5En%5Csum%5Climits_%7Bj%3D1%7D%5En%5Cleft(%5Csum%5Climits_%7Bk%3D1%7D%5E%7B%2B%5Cinfty%7D%5Cfrac%7B(-1)%5E%7Bk-1%7D(a_ia_j)%5Ek%7D%7Bk%7D%2B%5Csum%5Climits_%7Bk%3D1%7D%5E%7B%2B%5Cinfty%7D%5Cfrac%7B(a_ia_j)%5Ek%7D%7Bk%7D%5Cright)%5C%5C%0A%26%3D%5Csum%5Climits_%7Bi%3D1%7D%5En%5Csum%5Climits_%7Bj%3D1%7D%5En%5Csum%5Climits_%7Bk%3D1%7D%5E%7B%2B%5Cinfty%7D%5Cfrac%7B2a_i%5E%7B2k-1%7Da_j%5E%7B2k-1%7D%7D%7B2k-1%7D%5C%5C%0A%26%3D2%5Csum%5Climits_%7Bk%3D1%7D%5E%7B%2B%5Cinfty%7D%5Cfrac%201%7B2k-1%7D%5Csum%5Climits_%7Bi%3D1%7D%5En%5Csum%5Climits_%7Bj%3D1%7D%5En%20a_i%5E%7B2k-1%7Da_j%5E%7B2k-1%7D%5C%5C%0A%26%3D2%5Csum%5Climits_%7Bk%3D1%7D%5E%7B%2B%5Cinfty%7D%5Cfrac%201%7B2k-1%7D%5Cleft(%5Csum%5Climits_%7Bi%3D1%7D%5Ena_i%5E%7B2k-1%7D%5Cright)%20%5E2%5C%5C%0A%26%5Cgeqslant%200.%0A%5Cend%7Baligned%7D$

我们有 $%5Cprod%5Climits_%7Bi%3D1%7D%5En%5Cprod%5Climits_%7Bj%3D1%7D%5En%5Cfrac%7B1%2Ba_ia_j%7D%7B1-a_ia_j%7D%3De%5E%7B%5Cln%5Cprod%5Climits_%7Bi%3D1%7D%5En%5Cprod%5Climits_%7Bj%3D1%7D%5En%5Cfrac%7B1%2Ba_ia_j%7D%7B1-a_ia_j%7D%7D%5Cgeqslant%201.%5Cblacksquare$

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