Abstract
The main purpose of this paper is to investigate inequalities on symmetric sums of diagonalizable and positive definite tensors. In particular, we generalize the well-known Hlawka and Popoviciu inequalities to the case of diagonalizable and positive definite tensors. As corollaries, we extend Hlawka and Popoviciu inequalities for the combinatorial determinant, permanent and immanant of tensors, and generalized tensor functions.
Highlights
Let G be a subgroup of the symmetric group SI on the set {1, 2, . . . , I} and χ be an irreducible character of G
Macrcus and Minc [21] revealed a relationship between the generalized matrix function and a function involving the eigenvalues of normal matrices and considered the relationship between the generalized matrix function and an appropriate function of the singular values of an arbitrary square matrix
Chang et al [5] presented an inequality for Kronecker product of positive operators on Hilbert spaces and applied the inequality to generalized matrix functions
Summary
Let G be a subgroup of the symmetric group SI on the set {1, 2, . . . , I} and χ be an irreducible character of G. Huang et al [14] derived inequalities on no-integer power of products of generalized matrix functions on the sum of positive semi-definite matrices. Chang et al [5] presented an inequality for Kronecker product (sometimes called tensor product) of positive operators on Hilbert spaces and applied the inequality to generalized matrix functions. Paksoy et al [24] obtained some inequalities for generalized matrix functions of positive semi-definite matrices by an embedding and through kronecker products. Three different kinds of inequalities on generalized tensor functions associated with the diagonalizable and symmetric positive definite tensors are considered in Sections 4, 5 and 6, respectively
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