I'm trying to verify a bound for the gamma function $ \Gamma(z) = \int_0^\infty e^{-t}t^{z - 1}\;dt. $
In particular, for real $m \geq 1$, I'd like to show that $ \Gamma(m + 1) \leq 2\left(\frac{3m}{5}\right)^m. $
Knowing that the bound should be attainable, my first instinct is to split the integral as $ \Gamma(m + 1) = \int_0^{3m/5} e^{-t}t^{m}\;dt + \int_{3m/5}^\infty e^{-t}t^m\;dt \leq (1 - e^{-3m/5})\left(\frac{3m}{5}\right)^m + \int_{3m/5}^\infty e^{-t}t^m\;dt. $
Using integration by parts, $ \int_{3m/5}^\infty e^{-t}t^m\;dt = e^{-3m/5}\left(\frac{3m}{5}\right)^m + m\int_{3m/5}^\infty e^{-t}t^{m-1}\;dt.$
So the problem has been reduced to showing $ m\int_{3m/5}^\infty e^{-t}t^{m-1}\;dt \leq \left(\frac{3m}{5}\right)^m. $
But this doesn't seem to have made the problem any easier.
Any help is appreciated, thanks.