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Two proofs of the Kantorovich inequality and some generalizations. (English) Zbl 0656.60030

The Kantorovich inequality, \[ 1\leq (x'Ax)(x'A^{-1}x)\leq (a+b)^ 2/4ab \] where A is a positive-definite \(n\times n\) matrix with eigenvalues \(0<a=\lambda_ 1\leq \lambda_ 2\leq...\leq \lambda_ n=b\) and x is a unit vector, after diagonalizing, motivates the following two probability inequalities:
Let Z be a random variable taking values in [a,b], \(0<a<b<\infty\), then \[ 1\leq E(Z)E(1/Z)\leq (a+b)^ 2/4ab. \] Furthermore, if f is a convex function on [a,b], with \(f(a)=A>B=f(b)\), then \[ E(Z)Ef(Z)\leq \max \{(AB- Ba)^ 2/4(A-B)(b-a),\quad aA,\quad bB\}. \] At the same time, these two inequalities also give two probabilistic proofs of the Kantorovich inequality. Some further generalizations of these inequalities are discussed, too.
Reviewer: He Shengwu

MSC:

60E15 Inequalities; stochastic orderings
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