Let x, y be positive real number and m, n positive integers. The maximum value of the expression $$\dfrac{x^m y^n}{(1 + x^{2m}) (1 + y^{2n})}$$ is
$$\dfrac{x^my^n}{(1+x^{2m})(1+y^{2n)}}$$
Divide by $$x^my^n$$
$$=\dfrac{1}{\left( \dfrac{1}{x^m} + x^m\right)\left(\dfrac{1}{y^n}+y^n\right)}$$
by $$A.M\geq G.M$$
$$\dfrac{1}{x^m} + x^m \ge 2$$, $$\dfrac{1}{y^n} + y^n \ge 2$$
There by maximum value of $$\dfrac{x^my^n}{(1+x^{2m})(1+y^{2n})} = \dfrac{1}{2\times 2}=\dfrac{1}{4}$$
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