Calculate the product: $\prod\limits_{0<x<y<\infty} \left(x^{1/x} - y^{1/y}\right).$

Question

Calculate the product: $$\prod_{0<x<y<\infty} \left(x^{1/x} - y^{1/y}\right).$$

Answer 1

Well if you plot the graph $y=x^{1/x}$, you will notice that the graph reaches it's maxima at $x=e$ and $y=e^{1/e} \implies y\approx 1.4446$. So if you take any value of $y$ such that $y \gt 1 $, then $y^{1/y} \gt 1$. So $(x^{1/x} - y^{1/y}) \to 0$. Hence $$\prod_{0<x,y< \infty} (x^{1/x}-y^{1/y}) \to 0$$