It is well-known that for Hilbert space linear operators $0 leq A$ and $0 leq C$, inequality$C leq A$ does not imply $C^2 leq A^2.$ We introduce a strong order relation $0 leq B lll A$, which guarantees that $C^2 leq B^{1/2}AB^{1/2} text{for all} 0 leq C leq B,$ and that $C^2 leq A^2$ when $B$ commutes with $A$. Connections of this approach with the arithmetic-geometric mean inequality of Bhatia--Kittaneh as well as the Kantorovich constant of $A$ are mentioned.