Concerning polynomials on the unit interval

Abstract:

Let ${\mathcal {P}_n}$ be the normed linear space of all polynomials $p$ of degree $\leq n$ such that $p(1) = 0$ and $||p|| = (\int _{ - 1}^1 {|p(x){|^2}dx{)^{1/2}}}$. We determine sharp upper bounds for $|{a_n}|/||p||$ and $|{a_{n - 1}}|/||p||\;{\text {as}}\;p{\text {(x)}}\;{\text {: = }}\;\sum \nolimits _{\nu = 0}^n {{a_\nu }{x^\nu }}$ varies in ${\mathcal {P}_n}$.