In the real Hilbert space of self-adjoint elements of the tensor product ${\mathbb{M}}_m\otimes {\mathbb{M}}_n$, there are two natural cones besides the cone ${\mathfrak{P}}_0$ of positive semi-definite elements. The one is and the other is the cone ${\mathfrak{P}}_{-}$, dual to ${\mathfrak{P}}_{+}$ with respect to the inner product. Then, ${\mathfrak{P}}_{+}\subset {\mathfrak{P}}_0\subset {\mathfrak{P}}_{-}.$ A weak order relation $\ge$ is introduced by Our interest is in finding bounds for the ratio $|||T|||/|||S|||$ for $S\ge T\ge 0$, where $|||\cdot |||$ is one of the operator norm, the trace norm, and the Hilbert-Schmidt norm. $${\mathfrak{P}}_{+}:=\text{the convex hull of}\{X\otimes Y;0\le X\in {\mathbb{M}}_m,0\le Y\in {\mathbb{M}}_n\}$$ $$\mathbf{S}⪰\mathbf{T}\stackrel{def}{⟺}\mathbf{S}-\mathbf{T}\in {\mathfrak{P}}_{-}.$$