Inequality from Titu Andreescu

Algebra Level 4

Let P be a polynomial with positive co-efficient.if $P(1/x)\geq 1/P(x)$ holds for $x=1$ then

none of above it holds for every x>1 it holds for every x>0 it holds for every positive integer

1 solution

Abhishek Sinha
Oct 24, 2017

Let $P(x)= \sum_{n=0}^{N} a_n x^n$ , where $a_n >0$ . The given condition implies $P(1)^2 \geq 1$ . Now for any $x>0$ , we have $P(x)P(\frac{1}{x})= \big(\sum_{n=0}^{N} a_n x^n\big)\big( \sum_{n=0}^{N} a_n \frac{1}{x^n}\big) \stackrel{(a)}{\geq} (\sum_{n=0}^{N} a_n)^2 = P(1)^2 \geq 1,$ where the inequality (a) follows from positivity of the coefficients and an application of the Cauchy-Schwartz inequality.

Inequality from Titu Andreescu

1 solution

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