I have a question: I Can to proof that $\pi^e<e^{\pi}$ (it is easy), and also I proof that $\pi^{e^{\pi}}>e^{{\pi^e}}$ (by contradiction I assume that $\pi^{e^{\pi}}\leq e^{^{\pi^e}}$ and via natural logaritms obtains that $e^{\pi}<e^{\pi}\cdot \ln \pi <\pi^e$, an absurd whit the above inequality!)
But, the question is that I can not prove that $\pi^{e^{\pi^e}}<e^{\pi^{e^{\pi}}}$.
Any help!
Thanks!
