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

Abstract

Let P n ( x ) {P_n}(x) be an algebraic polynomial of degree ⩽ n \leqslant n having all its zeros inside [ − 1 , + 1 ] [ - 1, + 1] ; then we have \[ ∫ − 1 1 P n ′ 2 ( x ) d x > ( n / 2 ) ∫ − 1 1 P n 2 ( x ) d x . \int _{ - 1}^1 {P_n^{’2}(x)dx > (n/2)\int _{ - 1}^1 {P_n^2(x)dx.} } \] The result is essentially best possible. Other related results are also proved.