On Turánʹs inequality for Legendre polynomials

LetΔn(x)=Pn(x)2-Pn-1(x)Pn+1(x),where Pn is the Legendre polynomial of degree n. A classical result of Turán states that Δn(x) 0 for x [-1,1] and . Recently, Constantinescu improved this result. He established where hn denotes the nth harmonic number. We present the following refinement. Let n 1 be an integer. Then we have for all x [-1,1]αn(1-x2) Δn(x)with the best possible factorαn=μ[n/2]μ[(n+1)/2].Here, is the normalized binomial mid-coefficient.