双变量数列最值问题

aishuxue

求$\sqrt{2(2mn-1)}-\sqrt{(2m-1)(2n-1)}$,$m,n\in\mathbf{N^*}$的最小值.

aishuxue

没人会吗?

realnumber

$(2m-1)(2n-1)=4mn-2m-2n+1\le 4mn-4\sqrt{mn}+1$,设$t=\sqrt{mn}$
\[y=\sqrt{2(2mn-1)}-\sqrt{(2m-1)(2n-1)}\ge{\sqrt{2(2t^2-1)}-\sqrt{(2t-1)^2}}\]
\[=\sqrt{2(2t^2-1)}-2t+1=\frac{-2}{\sqrt{2(2t^2-1)}+2t}+1\]
可见t取最小，即t=1时，有最小值$\sqrt{2}-1$．