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Bound of $\frac{a_{n-1}^{2}}{a_{n-2}}$

Source: Macedonian TST 2022, P2

May 21, 2022
algebrapolynomialSymmetric inequalityinequalities

Problem Statement

Let n≥2n \geq 2 be a fixed positive integer and let a0,a1,...,an−1a_{0},a_{1},...,a_{n-1} be real numbers. Assume that all of the roots of the polynomial P(x)=xn+an−1xn−1+an−2xn−2+...+a1x+a0P(x) = x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+...+a_{1}x+a_{0} are strictly positive real numbers. Determine the smallest possible value of an−12an−2\frac{a_{n-1}^{2}}{a_{n-2}} over all such polynomials.
Proposed by Nikola Velov
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