Determinant Inequalities for Positive Definite Matrices Via Diananda’s Result for Arithmetic and Geometric Weighted Means

In this paper we prove among others that, if (A_j)_j=1,...,m are positive definite matrices of order n ≥ 2 and q_j ≥ 0, j = 1, ..., m with ∑j=1mqj=1$$\sum\nolimits_{j = 1}^m {{q_j} = 1} $$, then 0≤11−mini∈{1,…,m}{ qi }×[ ∑​i=1mqi(1−qi)[ det(Ai) ]−1−2n+1∑​1≤i<j≤mqiqj[ det(Ai+Aj) ]−1 ]≤∑​i=1mqi[ det(Ai) ]−1−[ det(∑​i=1mqiAi) ]−1≤1mini∈{1,…,m}{ qi }×[ ∑​i=1mqi(1−qi)[ det(Ai) ]−1−2n+1∑​1≤i<j≤mqiqj[ det(Ai+Aj) ]−1 ].