On Turán’s inequality for ultraspherical polynomials

Abstract

(0.3) An,x(x) -[F,,x(x)]2 Fn+l,,(x)Fn_,iX(x) > or 1, where Fn,x(x) =P(x)(x)/P((1). V. R. Thiruvenkatachar and T. S. Nanjundiah [4] and A. E. Danese [5] have obtained series of positive functions for Ai\(x) and its analogue in the case of the ultraspherical, Laguerre and Hermite polynomials. In the present paper we deepen the above results on ultraspherical polynomials by determining the signs of the first and second derivatives of AX(x) [P()(x) ]2 -P()1(x)Pn( I(x) for various values of X and x. We notice first of all that dA(?(x)/dx can be represented as a numerical multiple of the Wronskian Pn(+)1(x)dPn(? 1(x)/dx -p(?)(x)(dP(1(x)/dx) and express it also via the ChristoffelDarboux formula as a series of functions of constant sign. The sign of dA'?(x)/dx for various values of X and x follows easily from this. We determine the sign of (X 1)x(dA(?(x)/dx) for a general value of n and X and show that (d2/dx2)A() (x) 1 or 1/2?X<1, thus extending the result of B. S. Madhava Rao and V. R. Thiruvenkatachar to the case of ultraspherical polynomials. For integer values of n we obtain a series of positive functions for (d2/dx2)A(X) (x). Next we show that (1 x2)A4X) (x) decreases steadily in (0, 1) when 1<X <X3/2. From the signs of dA(?(x)/dx,