On Turán's inequality for Legendre polynomials

Abstract

Let Δ n ( x ) = P n ( x ) 2 - P n - 1 ( x ) P n + 1 ( x ) , where P n is the Legendre polynomial of degree n. A classical result of Turán states that Δ n ( x ) ⩾ 0 for x ∈ [ - 1 , 1 ] and n = 1 , 2 , 3 , … . Recently, Constantinescu improved this result. He established h n n ( n + 1 ) ( 1 - x 2 ) ⩽ Δ n ( x ) ( - 1 ⩽ x ⩽ 1 ; n = 1 , 2 , 3 , … ) , where h n denotes the nth harmonic number. We present the following refinement. Let n ⩾ 1 be an integer. Then we have for all x ∈ [ - 1 , 1 ] α n ( 1 - x 2 ) ⩽ Δ n ( x ) with the best possible factor α n = μ [ n / 2 ] μ [ ( n + 1 ) / 2 ] . Here, μ n = 2 - 2 n 2 n n is the normalized binomial mid-coefficient.