An analogue of some inequalities of P. Turán concerning algebraic polynomials having all zeros inside $[-1,+{}1]$. II

Abstract:

Let ${P_n}(x)$ be an algebraic polynomial of degree $\leqslant n$ having all zeros inside $[ - 1, + 1]$; then we have \[ \int _{ - 1}^1 {P’}_n^2 (x)dx > \left ( {\frac {n}{2} + \frac {3}{4} + \frac {3}{{4n}}} \right )\int _{ - 1}^1 {P_n^2(x)dx.} \] This bound is much sharper than found in [2]. Moreover, if ${P_n}(1) = {P_n}( - 1) = 0$, then under the above conditions we have \[ \int _{ - 1}^1 {P’}_n^2 (x)dx \geqslant \left ( {\frac {n}{2} + \frac {3}{4} + \frac {3}{{4(n - 1)}}} \right )\int _{ - 1}^1 {P_n^2(x)dx,} \] equality for ${P_n}(x) = {(1 - {x^2})^m},n = 2m$.