Dear all,

I need to calculate the minimum value of the following poynomial:

$\displaystyle P_{n}(\lambda):=\lambda(\lambda-1)\cdots(\lambda-n+1),$

where $\displaystyle n\in\mathbb{N}$ is even and $\displaystyle p\in\mathbb{R}$.

I can transform this polynomial into

$\displaystyle P_{n}(\lambda):=\lambda^{n}+\sum_{i=0}^{n-1}a_{i+1}\lambda^{n-1-i},$

by the means of Vieta's formula (PlanetMath: Vieta's formula) but this time it will not be easy to find the roots of $\displaystyle P_{n}^{\prime}=0$.

I really wonder if this is something well-known.

Thanks for the help again.

Note: It also helps me to find good upper and lower bounds for $\displaystyle \min\nolimits_{\lambda\in[0,(n-1)/2]}\big\{P_{n}(\lambda)\big\}$.