What are the minimum values?

Multiplying out and regrouping, the first expression is equal to $S_1 \; = \; \left[x + x^{-1} - 1009\right]^2 + \left[y +y^{-1} - 1009\right]^2 - 2\times1009^2 + 2\left(\sqrt{\tfrac{x}{y}} - \sqrt{\tfrac{y}{x}}\right)^2$ so we see that $S_1 \ge -2 \times 1009^2$ for all $x,y > 0$ , with equality when $x=y=\tfrac12\big(1009 + \sqrt{1009^2-4}\big)$ or $x=y=\tfrac12\big(1009 - \sqrt{1009^2-4}\big)$ . Similarly, the second expression is $S_2 \; = \; \left[x + x^{-1} + 1009\right]^2 + \left[y +y^{-1} + 1009\right]^2 - 2\times1009^2 + 2\left(\sqrt{\tfrac{x}{y}} - \sqrt{\tfrac{y}{x}}\right)^2$ Since $x + x^{-1} \ge 2$ for all $x > 0$ , it is clear that $S_2 \ge 2 \times 1011^2 - 2 \times 1009^2$ for all $x,y > 0$ , with equality when $x=y=1$ . This makes the desired answer $2\big(1011^2 - 2\times1009^2\big) \; =\; \boxed{-2028082}$