Consider the following inequality of Lemma 1 arising in The law of large numbers for quantum stochastic filtering and control of many-particle systems :
$$\Big|tr(L\gamma LB) - \frac{1}{2}tr(B(L\gamma + \gamma L))tr(BL + \gamma L) + tr(B\gamma)tr(BL)tr(\gamma L) \Big| \leq 5||L||^2 tr\big((Id-\gamma)B\big),$$
where $\gamma$ is a ($n\times n$) density matrix of rank one (see https://en.wikipedia.org/wiki/Density_matrix for the definition of density matrix), $B$ is a density matrix and $L$ a hermitian matrix. My question is whether the above inequality holds (up to some constant in front of $tr\big((Id-\gamma)B\big)$) by assuming only that $\gamma$ is a density matrix (with the assumptions on $\gamma, L$ unchanged)?
PS : Without loss of generality we may assume that $\gamma$ is diagonal with its diagonal elements $\gamma_1,\ldots, \gamma_n\in\mathbb R_+$ s.t. $\sum_{i=1}^N \gamma_i=1$.
PS2 : Denote respectively the l.h.s. and r.h.s. by
$$f(\gamma):=\Big|tr(L\gamma LB) - \frac{1}{2}tr(B(L\gamma + \gamma L))tr(BL + \gamma L) + tr(B\gamma)tr(BL)tr(\gamma L) \Big|$$
and
$$g(\gamma):=5||L||^2 tr\big((Id-\gamma)B\big).$$
Following the hint of Narutaka OZAWA, I try to reduce the general case to the case where $\gamma$ has rank one. The difficulty for me is that $g$ is affine while $f$ is not (it is even not trivial for me to show $f$ is convex).
