Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Higher Order Turán Inequalities

Author: Dimitar K. Dimitrov
Journal: Proc. Amer. Math. Soc. 126 (1998), 2033-2037
MSC (1991): Primary 30D10, 33C45
MathSciNet review: 1459117
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The celebrated Turán inequalities $P_{n}^{2}(x) - P_{n-1}(x) P_{n+1}(x) \geq 0, \ \ x \in [-1,1],\ \ n \geq 1$, where $P_{n}(x)$ 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 $\gamma _{n}^{2} - \gamma _{n-1} \gamma _{n+1} \geq 0,\ \ n \geq 1$, which hold for the Maclaurin coefficients of the real entire function $\psi$ in the Laguerre-Pólya class, $\psi(x) = \sum _{n=0}^{\infty} \gamma _{n} x^{n}/n!$.

References [Enhancements On Off] (What's this?)

  • 1. R. Boas, Entire functions, Academic Press, New York, 1954. MR 16:914f
  • 2. T. Craven and G. Csordas, Jensen polynomials and the Turán and Laguerre inequalities, Pacific J. Math. 136(1989), 241-260. MR 90a:26035
  • 3. G. Csordas, T. S. Norfolk and R. S. Varga, The Riemann hypothesis and the Turán inequalities, Trans. Amer. Math. Soc. 296(1986), 521-541. MR 87i:11109
  • 4. G. Csordas and R. S. Varga, Necessary and sufficient conditions and the Riemann hypothesis, Adv Appl. Math. 11(1990), 328-357. MR 91d:11107
  • 5. J. Dombrowski, Spectral properties of phase operators, J. Math. Phys. 15(1974), 576-577. MR 48:13075
  • 6. J. Dombrowski, Tridiagonal matrix representations of cyclic self-adjoint operators, I, II, Pacific J. Math. 114(1984), 325-334; 120(1985), 47-53. MR 87m:47046; MR 85h:47033
  • 7. J. Dombrowski and P. Nevai, Orthogonal polynomials, measures and recurrence relations, SIAM J. Math. Anal. 17(1986), 752-759. MR 87g:42039
  • 8. J. S. Geronimo and W. Van Assche, Approximating the weight function on several intervals, J.Approx.Theory 65(1991), 341-371. MR 93b:42037
  • 9. G. Gasper, An inequality of Turán for Jacobi polynomials, Proc. Amer. Math. Soc. 32(1972), 435-439. MR 44:7013
  • 10. S. Karlin and G. Szeg\H{o}, On certain determinants whose elements are orthogonal polynomials, J. Analyse Math. 8(1960/61), 1-157; reprinted in: G. Szeg\H{o}: collected papers, Vol. 3, (R.Askey, Ed.), Birkhäuser, Boston, 1982, pp. 605-773. MR 26:539
  • 11. J. Ma$\check{r}$ík, On polynomials with all real zeros, $\check{C}$asopis P$\check{e}$st. Mat. 89(1964), 5-9. MR 31:4782
  • 12. A. Máté, P. Nevai and V. Totik, Asymptotics for orthogonal polynomials defined by a recurrence relation, Constr. Approx. 1(1985), 231-284. MR 89a:42035
  • 13. G.V. Milovanovi\'{c}, D.S. Mitrinovi\'{c} and Th. M. Rassias, Topics in polynomials: extremal problems, inequalities, zeros, World Scientific, Singapore, 1994. MR 95m:30009
  • 14. N. Obrechkoff, Zeros of polynomials, Publ. Bulg. Acad. Sci., Sofia, 1963.(in Bulgarian); German translation: Verteilung und Berchnung der Nullstellen Reeller Polynome, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963. MR 29:1303
  • 15. G. Pólya, Über die algebraisch-funktionentheoretischen Untersichungen von J. L. W. V. Jensen, Kgl. Danske Vid. Sel. Math.-Fys. Medd. 7 (1927), 3-33.
  • 16. G. Pólya and J. Schur, Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen, J. Reine Angew. Math. 144(1944), 89-113.
  • 17. B. Riemann, Über die Anzahl der Primzahlen unter einen gegebenen Grösse, Monatsh. Der Berliner Akad. (1858/60), 671-680; also in: Gesammelte Mathematische Werke, 2nd edition, Teubner, Leipzig, 1982, No. VII, pp. 145-153.
  • 18. G. Szeg\H{o}, On an inequality of P. Turán concerning Legendre polynomials, Bull. Amer. Math. Soc. 54 (1948), 401-405; reprinted in: G. Szeg\H{o}: Collected Papers, Vol. 3, (R.Askey, ed.), Birkhäuser, Boston, 1982, pp. 69-73. MR 9:429d; MR 84d:01082c
  • 19. W. Van Assche and J. S. Geronimo, Asymptotics for orthogonal polynomials on and off the essential spectrum, J.Approx. Theory 55(1988), 220-231. MR 89m:33022
  • 20. R. S. Varga, Scientific computation on mathematical problems and conjectures, Regional Conf. Ser. Appl. Math. Vol. 60, SIAM, Philadelphia, PA, 1990. MR 92b:65012

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 30D10, 33C45

Retrieve articles in all journals with MSC (1991): 30D10, 33C45

Additional Information

Dimitar K. Dimitrov
Affiliation: Departamento de Ciências de Computação e Estatística, IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP, Brazil

Keywords: Tur\'{a}n inequalities, Tur\'{a}n determinants, entire functions in the Laguerre-P\'{o}lya class, Riemann hypothesis
Received by editor(s): December 12, 1996
Additional Notes: Research supported by the Brazilian foundation CNPq under Grant 300645/95-3 and the Bulgarian Science Foundation under Grant MM-414.
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society