Let $A,B\geq0$ be positive semidefinite matrices of arbitrary size $n\times n$. Denote by $\alpha$ and $\beta$ their largest eigenvalues.

It is well-known that the eigenvalues of the expression $AB + BA$ are bounded by [F. Zhang, "Matrix Theory, Sec. 7.2]

$$ -\frac{1}{4}\alpha\beta I \,\,\leq\,\, AB + BA \,\,\leq\,\, 2\alpha\beta I. $$

I can show that $$ - \rm{tr}(AB)I - \rm{tr}(A)B - \rm{tr}(B)A \,\,\leq\,\, AB + BA \,\,\leq\,\, \rm{tr}(AB)I + \rm{tr}(B)A + \rm{tr}(A)B. $$

Now normalize to $\rm{tr}(A) = \rm{tr}(B) = 1$. When $\rm{tr}(AB) = 0$, that is $A\perp B$ in terms of the Hilbert-Schmidt inner product, the expression reduces to

$$ -(A+B) \,\,\leq\,\, AB + BA \,\,\leq\,\, (A + B). $$

These inequalities look rather simple and are for $n\geq 3$ even tight. However I failed to find them in the literature. Are these known, or can they straightforwardly be derived from other expressions?