Two determinantal inequalities for positive semidefinite matrices

Let A,B,C ∈ Cnxn be positive semidefinite matrices. In this paper, the authors prove two determinantal inequalities|A+B+C|+|C|≥|A+C|+|B+C|+(3n −2 n+1+1)|ABC|1/3and|A+B+C|+|A|+|B|+|C|≥|A+B|+|A+C|+|B+C|+3(3n−1−2n+1)|ABC|1/3.These two inequalities improve known ones.

Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices

In this paper, the authors extend determinantal inequalities of the Hua-Marcus-Zhang type for positive definite matrices to the corresponding ones for quaternion matrices.

Weak Log-majorization of Unital Trace-preserving Completely Positive Maps

Let Φ : Mn → Mn be a unital trace preserving completely positive map and A ∈ Mn be a positive definite matrix. Weak log-majorization and weak majorization between Φ(A) and A are studied. Determinantal inequalities between Φ(A) and A are obtained as a consequence. By considering special classes of unital trace preserving completely positive map, some known matrix inequalities such as Fischer’s inequality are rediscovered. An affirmative answer to a question of Tam and Zhang in 2019 is given.

Determinantal inequalities for the partition function

Let p(n) denote the partition function. In this paper, we will prove that for $n\ges 222$, $$\left| {\matrix{ {p(n)} & {p(n + 1)} & {p(n + 2)} \cr {p(n-1)} & {p(n)} & {p(n + 1)} \cr {p(n-2)} & {p(n-1)} & {p(n)} \cr } } \right| > 0.{\rm }$$As a corollary, we deduce that p(n) satisfies the double Turán inequalities, that is, for $n\ges 222$, $$(p(n)^2-p(n-1)p(n+1))^2-(p(n-1)^2-p(n-2)p(n))(p(n+1)^2-p(n)p(n+2))>0.$$