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101 (2010), 73–104. J. Borcea and P. Brändén, Hyperbolicity preservers and majorization, C. R. Math. Acad. Sci. Paris 348 (2010), 843–846. J. Borcea and P. Brändén, The Lee–Yang and Pólya–Schur programs. II. Theory of stable poly­ nomials and applications, to appear in Comm. Pure Appl. Math. 3087. P. Brändén and E. Ottergren, A characterization of multiplier sequences for generalized La­ guerre bases, arXiv: 0581078. P. Brändén, Iterated sequences and the geometry of zeros, J. Reine Angew. Math.

Our next theorems show that the normalized moments of admissible kernels satisfy the Turán inequalities. 34 | George Csordas Theorem 23 (G. Csordas and R. S. Varga [45]). Let ???? : ℝ ????→ ℝ be an admissible ker­ nel. Let ∞ ∞ ???? ????(????) := ∫ ????(????) cos(????????) ???????? = ∑ ???? (−????2 )???? , (8) (2????)! ????=0 0 where the moments are defined by ∞ ???????? := ∫ ????2???? ????(????) ???????? (???? ≥ 0). 0 If log(????(√????)) is concave on (0, ∞), then the Turán inequalities hold ( for ???? ≥ 1), ????????2 − ????????−1 ????????+1 ≥ 0, where ???????? := ????!

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