Problem 6

For real numbers

x_1,x_2,y_1,y_2,z_1,z_2

satisfying

x_1>0

x_2>0

x_1y_1>z_1^2

, and

x_2y_2>z_2^2

, prove that

\frac{8}{(x_1+x_2)(y_1+y_2)-(z_1+z_2)^2}\le\frac1{x_1y_1-z_1^2}+\frac1{x_2y_2-z_2^2}

. Give necessary and sufficient conditions for equality.

Step 5 of 8: Combine the cross estimates

2\sqrt{a_1a_2}\le x_1y_2+x_2y_1-2z_1z_2

Detailed analysis

Subtract the square estimate from the weighted AM-GM estimate. Using $a_1=x_1y_1-z_1^2$ and $a_2=x_2y_2-z_2^2$ yields the displayed cross-term bound.