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In the book Linear Integral Equations by Rainer Kress at page 138 is stated without giving a proof this inequality:

$$v^\delta e^{-v}\leq \delta^\delta e^{-\delta},$$ for all $v,\delta\in]0,\infty[$. I've tried to prove it, but I haven't succeed. Any ideas?

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One has $\ln(x)\leq x-1$ for all positive $x$, and so in particular for $x=\frac{v}{\delta}$. Playing around with $\ln(\frac{v}{\delta})\leq \frac{v}{\delta}-1$ results in your inequality.

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    $\begingroup$ maybe you mean $ln(x)\leq x-1$? $\endgroup$ – rafforaffo Feb 3 '15 at 16:18
  • $\begingroup$ @rafforaffo oops thanks. $\endgroup$ – Casteels Feb 3 '15 at 16:20

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