By Klin M., et al.

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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 ???????? := ????!

7 (1950), 197–226. D. A. Cardon, Turán inequalities and subtraction-free expressions, JIPAM. J. Inequal. Pure Appl. Math. Article 91 9 (2008), 11pp. D. A. Cardon, Extended Laguerre inequalities and a criterion for real zeros, in: Progress in Anal ysis and its Applications, pp. 143–149, World Sci. , Hackensack, NJ, 2010. J. B. Conrey and A. Ghosh, Turán inequalities and zeros of Dirichlet series associated with cer tain cusp forms, Trans. Amer. Math. Soc. 342 (1994), 407–419. M. Chasse, Linear Preservers and Entire Functions with Restricted Zero Loci, Ph.

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