I have considered a function $f (x) = x^{x+1}-(x+1)^x$ , with $f : [2022;+\infty)\to R$ .
I have made the signs table for the derivative of the function, assuming x on the respective interval. The sign of the derivative is positive on the interval $[2022;+\infty)$, which means the function is increasing.
I have calculated $f(2022)$ which is $2022^{2023}-2023^{2022}$ and $\displaystyle\lim_{x\to\infty}f(x)$, which is $\infty$.
In conclusion, I found the monotony for the function, but still haven't proved the inequality.
Any serious answear is appreciated.
