Some inequalities of algebraic polynomials having all zeros inside [-1,1

Abstract

Let H n {H_n} be the set of all algebraic polynomials whose degree is n n and whose zero are all real and lie inside [ − 1 , 1 ] [ - 1,1] . Then for n n even we have ( n = 2 m ) (n = 2m) \[ ∫ − 1 1 ( P n ( x ) ) 2 ⩾ ( n / 2 + 3 / 4 + 3 / 4 ( n − 1 ) ) ∫ − 1 1 P n 2 ( x ) d x , \int _{ - 1}^1 {{{({P_n}(x))}^2} \geqslant (n/2 + 3/4 + 3/4(n - 1))\int _{ - 1}^1 {P_n^2(x)dx} } , \] where equality holds iff P n ( x ) = ( 1 − x 2 ) m {P_n}(x) = {(1 - {x^2})^m} . If n n is an odd positive integer, a similar inequality is valid (see (1.6) below). In the case P n ∈ H n {P_n} \in {H_n} and subject to the condition P n ( 1 ) = 1 {P_n}(1) = 1 , then \[ ∫ − 1 1 ( P ′ n ( x ) ) 2 d x ⩾ n 4 + 1 8 + 1 8 ( 2 n − 1 ) , \int _{ - 1}^1 {({{P’}_n}} (x){)^2}dx \geqslant \frac {n}{4} + \frac {1}{8} + \frac {1}{{8(2n - 1)}}, \] , where equality holds for P n ( x ) = ( ( 1 + x ) / 2 ) n {P_n}(x) = {((1 + x)/2)^n} .