TITLE: Integral inequality $\int_{0}^{e}\operatorname{W(x)^{\pi}}>1$
QUESTION [7 upvotes]: Hi it's a problem of mine :


Prove that (without calculating the integral) :
    $$\int_{0}^{e}\operatorname{W(x)^{\pi}}\mathrm{d}x>1$$
    Where $\operatorname{W(x)}$ denotes the Lambert's function .


I have tested during 2 hours some methods but it fails always because the inequality is pretty sharp .Particulary I have used the power series of the Lambert's function but it's not convincing . So I really need help on this question .
If you have nice ideas...
...Thanks a lot.

REPLY [2 votes]: Not very rigorous.
Use the Taylor series of $W(x)$ around $x=e$ up to order $O\left((x-e)^{n+1}\right)$, raise it to power $\pi$ to get things like
$$1+\frac{\pi  (x-e)}{2 e}+\frac{\pi  (2 \pi -5) (x-e)^2}{16 e^2}+\frac{\pi 
   \left(45-30 \pi +4 \pi ^2\right) (x-e)^3}{192 e^3}+\frac{\pi  (-583+510 \pi -120 \pi ^2+8 \pi ^3) (x-e)^4}{3072 e^4}+O\left((x-e)^5\right)$$ Integrate termwise between the bounds to get $\big[e\times P_n(\pi)\big]$
Below are given the decimal representation of the results
$$\left(
\begin{array}{cc}
n & \text{result} \\
 1 & 0.58334827279065 \\
 2 & 0.81164122032718 \\
 3 & 0.92027110969382 \\
 4 & 0.96636673732565 \\
 5 & 0.98665793725515  \\
 6 & 0.99583123255972  \\
 7 & 0.99996493384791  \\
 8 & 1.00173198491504 \\
 9 & 1.00236972369043 \\
 10 & 1.00247298096855 \\
 11 & 1.00233463559724 \\
 12 & 1.00210020862900 \\
 13 & 1.00184132446202 \\
 14 & 1.00159192031259 \\
 15 & 1.00136660705921 \\
 16 & 1.00117017261118 \\
 17 & 1.00100256311893 \\
 18 & 1.00086150355766 \\
 19 & 1.00074386588970 \\
 20 & 1.00064636610536
\end{array}
\right)$$