I am currently reading a treatment on Fourier Series and the Heat trace, where the following estimate is stated:
Suppose $\lambda > 0$ and $t \in (0,\infty)$. We have
\begin{equation} e^{-t\lambda}\lambda^j \leq t^{-j} C(j) e^{-t\lambda/2} \end{equation} where $C(j)$ denotes a constant, dependent on the positive number $j$.
I'd like to understand how to derive this, but struggle to fill in the details - do I need to expand the exponential into its power series expression to see this ?
Many thanks for your help !