Zhan's inequality on A-G mean inequalities

Let A, B, X be n×n matrices such that A, B are positive definite, and f, g be two continuous functions on (0,∞), k=max⁡{f(λ)g(λ)λ|λ∈σ(A)∪σ(B)}, where σ(A) is the spectrum of A. If f(x)g(x) is Kwong, i.e., the matrix (fg(λi)+fg(λj)λi+λj) is positive semidefinite for any positive numbers λ1,⋯,λn, then for any β>0,i.⦀A12(f(A)Xg(B)+g(A)Xf(B))B12⦀≤k⦀2(1−2β+2βr0)AXB+4β(1−r0)t+2(A2X+tAXB+XB2)⦀ holds for −2