I found the following result, When i was working on my calculator . $$x^y < y^x \quad ,x < y \quad \text{ for } x,y<e$$ $$x^y > y^x \quad ,x < y \quad \text{ for } x,y>e$$
I can't find a proof without convergence
2026-04-13 03:13:21.1776050001
Relation of $e$ to other numbers...
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Hint: Take the logarithm: $y\ln x<x\ln y$ and consider the function $f(x)=\frac{\ln x}{x}$.