Determinant Inequalities for Positive Definite Matrices Via Additive and Multiplicative Young Inequalities

En este trabajo demostramos entre otros que, si las matrices definidas positivas A, B de orden n satisface la condiciÃ3n0 < mIn ≤ B −A ≤ M In,para algunas constantes 0 < m < M, donde In es la matriz identidad, entonces0 ≤ (1 − t) [det (A)]−1 + t [det (A + mIn)]−1 − [det (A + mtIn)]−1≤ (1 − t) [det (A)]−1 + t [det (B)]−1 − [det ((1 − t) A + tB)]−1≤ (1 − t) [det (A)]−1 + t [det (A + M In)]−1 − [det (A + M tIn)]−1 ,para todo t ∈ [0, 1]In this paper we prove among others that, if the positive definite matrices A, B of order n satisfy the condition 0 < mIn ≤ B − A ≤ M In, for some constants 0 < m < M, where In is the identity matrix, then0 ≤ (1 − t) [det (A)]−1 + t [det (A + mIn)]−1 − [det (A + mtIn)]−1≤ (1 − t) [det (A)]−1 + t [det (B)]−1 − [det ((1 − t) A + tB)]−1≤ (1 − t) [det (A)]−1 + t [det (A + M In)]−1 − [det (A + M tIn)]−1 ,for all t ∈ [0, 1