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

Abstract

Abstract In this paper we prove among others that, if (Aj ) j=1,...,m are positive definite matrices of order n ≥ 2 and qj ≥ 0, j = 1, ..., m with ∑ j = 1 m q j = 1 $$\sum\nolimits_{j = 1}^m {{q_j} = 1} $$ , then 0 ≤ 1 1 − min i ∈ { 1 , … , m } { q i } × [ ∑ ​ i = 1 m q i ( 1 − q i ) [ det ( A i ) ] − 1 − 2 n + 1 ∑ ​ 1 ≤ i < j ≤ m q i q j [ det ( A i + A j ) ] − 1 ] ≤ ∑ ​ i = 1 m q i [ det ( A i ) ] − 1 − [ det ( ∑ ​ i = 1 m q i A i ) ] − 1 ≤ 1 min i ∈ { 1 , … , m } { q i } × [ ∑ ​ i = 1 m q i ( 1 − q i ) [ det ( A i ) ] − 1 − 2 n + 1 ∑ ​ 1 ≤ i < j ≤ m q i q j [ det ( A i + A j ) ] − 1 ] .