This note aims to prove a very interesting property concerning the monotonicity of trace inequalities of square matrices.
Before going further, let us denote by the set (or the algebra) of complex square matrices. Then, by we mean the set of Hermitian matrices in . We also denote by the set of positive semi-definite matrices in . In other words, there holds
As usual, the notation means for any .
The following inequality is basically due to Hoa-Tikhonov [here].
Theorem 1. Let and let a function be Borel measurable. The inequality
holds for all with if and only if the function is convex on .
The proof of the above theorem is rather simple but elegant. The idea is to transform the condition into the relation for some with . Then the theorem follows immediately from the Jensen trace inequality for contractions.
It is also interesting to note that the super-additivity property, i.e.
is equivalent to the convexity of the function .
The question of determining whether some special cases of the above inequality characterize scalar multiples of the trace among all positive linear functions. First, Hoa and Tikhonov also prove the following result.
Theorem 2. Let a function be Borel measurable and let a positive linear function on be such that
whenever . Then either is constant on or is a scalar multiple of the trace.
They then prove that
Theorem 3. Let be unequal positive numbers. If for a positive linear functional on , the inequality
holds for all , then is a scalar multiple of the trace.
This result immediately leads to the following
Theorem 4. Let be a positive number. Let a positive linear functional on be such that
whenever . Then is a scalar multiple of the trace.
It is clear to see that Theorem 4 includes a particular case the following well-known result: If
for all satisfying , then is a scalar multiple of the trace.