Now i know all parameters are positive,i mean, $g_{AB},g_{A},g_{AC},P_{B2AC},P_{C2AB},P_{A2BC},\mu$ and $N_0$ are all > $0$,and the only variable $\rho^{st}_A \in [0,1].$
I think the formula i mention above is convex,because its differentiation is larger than $0$,is my thinking right? or should i consider about other condition?