A proof of Bellman problem (Q2739057)

The Bellman problem concerns the arithmetic-geometric inequality for traces of positive semidefinite matrices. More precisely, it asserts that if \(A_1,\dots,A_n\) are \(n\) positive semidefinite matrices, then NEWLINE\[NEWLINE|\text{tr} (A_1, \dots,A_n)|\leq{1\over n}\bigl(\text{tr}(A^n_1)+ \cdots +\text{tr}(A^n_n) \bigr).NEWLINE\]NEWLINE This note offers a somewhat simple proof, which is built step-by-step from the case of \(n=2\).NEWLINENEWLINENEWLINEThere is another version of the Bellman inequality which appeared in the literature. For \(n=2\), this says that NEWLINE\[NEWLINE\bigl(\text{tr} (A_1A_2) \bigr)^{1/2} \leq(\text{tr} A_1+\text{tr} A_2)/2NEWLINE\]NEWLINE for positive semidefinite \(A_1\) and \(A_2\) (matrices or operators on a Hilbert space). This can be found in the paper of \textit{H. Neudecker} [J. Math. Anal. Appl. 166, No. 1, 302-303 (1992; Zbl 0760.15015)] and the references therein.