This question is related to topics in quantum information but I will present it as a linear algebra question here.
Consider some matrix $\delta_{AB}$ that lives in a bipartite Hilbert space $H_A\otimes H_B$. The partial trace map traces over one of the Hilbert space so we obtain the reduced matrices as below
$$\delta_A = \text{Tr}_B(\delta_{AB}), \, \, \delta_B = \text{Tr}_A(\delta_{AB})$$
Denote $A\geq B$ to mean that $A-B$ is positive semidefinite. Let $\rho_{AB}$ and $\sigma_{AB}$ both be positive semidefinite matrices with trace $1$. Let $\lambda, \lambda_1, \lambda_2$ each be the smallest real numbers such that the following relationships hold
$$\rho_{AB} - \lambda\sigma_{AB} \geq 0$$ $$\rho_{A} - {\lambda_1}\sigma_{A} \geq 0$$ $$\rho_{B} - {\lambda_2}\sigma_{B} \geq 0$$
What is the relationship between $\lambda$, $\lambda_1$ and $\lambda_2$? I'm looking for some kind of non-trivial inequality that relates all three.
By partial tracing over the conditions above, one sees that $\lambda\geq \lambda_1$ and similarly $\lambda\geq\lambda_2$ but I feel that more can be said.