On Turán's 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 n=1,2,3,…. Recently, Constantinescu improved this result. He establishedhnn(n+1)(1-x2)⩽Δn(x)(-1⩽x⩽1;n=1,2,3,…),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, μn=2-2n2nn is the normalized binomial mid-coefficient.