Some inequalities of algebraic polynomials having real zeros

Abstract:

Let ${P_n}(x)$ be an algebraic polynomial of degree n having all real zeros. We set \[ {I_n} = \frac {{{{\left \| {{{P’}_n}(x)\omega (x)} \right \|}_{{L_2}[a,b]}}}}{{{{\left \| {{P_n}(x)\omega (x)} \right \|}_{{L_2}[a,b]}}}}.\] In this work the lower and upper bounds of ${I_n}$ are investigated under the assumptions that all the zeros of ${P_n}(x)$ are inside $[a,b]$ and outside $[a,b]$, respectively. We restrict ourselves here with two cases, (1) $\omega (x) = {(1 - {x^2})^{1/2}},[a,b] = [ - 1,1]$; (2) $\omega (x) = {e^{ - x/2}},[a,b] = [0,\infty )$. Results are shown to be best possible.