Higher order Turán inequalities

Abstract

Abstract. The celebrated Turán inequalitieswhere Pn(x) denotes the Legendre polynomial of degree n, are extended to inequalities for sums of products of four classical orthogonal polynomials. The proof is based on an extension of the inequalities γ 2 n − γ n−1 γ n+1 ≥ 0, n ≥ 1, which hold for the Maclaurin coefficients of the real entire function ψ in the Laguerre-Pólya class, ψ(x) = ∞ n=0 γnx n /n!.

Differential equation and inequalities of the generalized k-Bessel functions

“…Turán [ 30 ] proved that the Legendre polynomials satisfy the determinantal inequality where , and the equality occurs only for . The inequalities similar to ( 1.10 ) can be found in the literature [ 2 , 3 , 5 , 11 , 16 , 25 ] for several other functions, for example, ultraspherical polynomials, Laguerre and Hermite polynomials, Bessel functions of the first kind, modified Bessel functions, and the polygamma function. Karlin and Szegö [ 24 ] named determinants in ( 1.10 ) as Turánians.…”

Turán Inequalities and Zeros of Orthogonal Polynomials

“…For symmetric polynomials (i.e. when p k (−x) = (−1) k p k (x)) and the monic normalization, the case we mainly deal with in this paper, the recurrence (1) can be rewritten as (5) p k+1 (x) = xp k (x) − c k p k−1 (x), and (3), ( 4) become (6) x kk = −x 1k = 2 max k−2 i=0…”

“…In the following theorem the first part belongs to M. Patrick [19] and the second one to J. Maříc [16] (the extension of it to the whole Laguerre-Pólya class is due to D.K. Dimitrov [5]).…”

“…Theorem 6. (i) Let p k be a symmetric polynomial satisfying (5) and suppose that c k are nondecreasing. Then…”

“…Bounds from higher order Turán inequalities. In this section we will consider only the symmetric case (5). For convenience we put c 0 = 0.…”