Let x, y be positive real numbers and m, n positive integers. The maximum value of the expression (x^m y^n)/((1+x^2 m )(1+y^2 n )) is:

Let $\displaystyle x, y$ be positive real numbers and $\displaystyle \mathrm{m}, \mathrm{n}$ positive integers. The maximum value of the expression $\displaystyle \frac{x^{\mathrm{m}} y^{\mathrm{n}}}{\left(1+x^{2 \mathrm{~m}}\right)\left(1+y^{2 \mathrm{n}}\right)}$ is: