An inequality for Jacobi polynomials: a complement to Finite Increment Theorem

Let P=Pₙ(α,β) be the n-th degree Jacobi polynomial,which is orthogonal in $[-1,1]$ with respect to the weight function(1-x)α(1+x)β, α,β>-1. For parameters(α,β) satisfying either α≥β≥ 1/2 orα≥ 1/2, β=-1/2, we prove the inequalityP(1)-P(x)≥ P^(x)\,(1-x), x∈ [0,1],which may be viewed as a complement to Finite Increment Theorem forJacobi polynomials.