Higher order Turán inequalities
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- by Dimitar K. Dimitrov PDF
- Proc. Amer. Math. Soc. 126 (1998), 2033-2037 Request permission
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
- Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
- Thomas Craven and George Csordas, Jensen polynomials and the Turán and Laguerre inequalities, Pacific J. Math. 136 (1989), no. 2, 241–260. MR 978613, DOI 10.2140/pjm.1989.136.241
- George Csordas, Timothy S. Norfolk, and Richard S. Varga, The Riemann hypothesis and the Turán inequalities, Trans. Amer. Math. Soc. 296 (1986), no. 2, 521–541. MR 846596, DOI 10.1090/S0002-9947-1986-0846596-4
- George Csordas and Richard S. Varga, Necessary and sufficient conditions and the Riemann hypothesis, Adv. in Appl. Math. 11 (1990), no. 3, 328–357. MR 1061423, DOI 10.1016/0196-8858(90)90013-O
- J. Dombrowski, Spectral properties of phase operators, J. Mathematical Phys. 15 (1974), 576–577. MR 334757, DOI 10.1063/1.1666686
- Peng Fan, Remark on: “Tridiagonal matrix representations of cyclic selfadjoint operators” [Pacific J. Math. 114 (1984), no. 2, 325–334; MR0757504 (85h:47033)] by J. Dombrowski, Proc. Amer. Math. Soc. 98 (1986), no. 1, 85–88. MR 848881, DOI 10.1090/S0002-9939-1986-0848881-4
- Joanne Dombrowski and Paul Nevai, Orthogonal polynomials, measures and recurrence relations, SIAM J. Math. Anal. 17 (1986), no. 3, 752–759. MR 838253, DOI 10.1137/0517054
- J. S. Geronimo and W. Van Assche, Approximating the weight function for orthogonal polynomials on several intervals, J. Approx. Theory 65 (1991), no. 3, 341–371. MR 1109412, DOI 10.1016/0021-9045(91)90096-S
- George Gasper, An inequality of Turán type for Jacobi polynomials, Proc. Amer. Math. Soc. 32 (1972), 435–439. MR 289826, DOI 10.1090/S0002-9939-1972-0289826-8
- S. Karlin and G. Szegö, On certain determinants whose elements are orthogonal polynomials, J. Analyse Math. 8 (1960/61), 1–157. MR 142972, DOI 10.1007/BF02786848
- Jan Mařík, On polynomials, all of whose zeros are real, C̆asopis Pěst. Mat. 89 (1964), 5–9 (Czech, with Russian and German summaries). MR 0180548, DOI 10.21136/CPM.1964.117490
- Attila Máté, Paul Nevai, and Vilmos Totik, Asymptotics for orthogonal polynomials defined by a recurrence relation, Constr. Approx. 1 (1985), no. 3, 231–248. MR 891530, DOI 10.1007/BF01890033
- G. V. Milovanović, D. S. Mitrinović, and Th. M. Rassias, Topics in polynomials: extremal problems, inequalities, zeros, World Scientific Publishing Co., Inc., River Edge, NJ, 1994. MR 1298187, DOI 10.1142/1284
- Nikola Obreškov, Nuli na polinomite, Izdat. BЪlgar. Akad. Nauk., Sofia, 1963 (Russian). MR 0164004
- G. Pólya, Über die algebraisch-funktionentheoretischen Untersichungen von J. L. W. V. Jensen, Kgl. Danske Vid. Sel. Math.-Fys. Medd. 7 (1927), 3-33.
- G. Pólya and J. Schur, Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen, J. Reine Angew. Math. 144(1944), 89-113.
- 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.
- Garrett Birkhoff and Morgan Ward, A characterization of Boolean algebras, Ann. of Math. (2) 40 (1939), 609–610. MR 9, DOI 10.2307/1968945
- Walter Van Assche and Jeffrey S. Geronimo, Asymptotics for orthogonal polynomials on and off the essential spectrum, J. Approx. Theory 55 (1988), no. 2, 220–231. MR 965218, DOI 10.1016/0021-9045(88)90088-3
- Richard S. Varga, Scientific computation on mathematical problems and conjectures, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 60, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990. MR 1068317, DOI 10.1137/1.9781611970111
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
- MR Author ID: 308699
- Email: dimitrov@nimitz.dcce.ibilce.unesp.br
- 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
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 2033-2037
- MSC (1991): Primary 30D10, 33C45
- DOI: https://doi.org/10.1090/S0002-9939-98-04438-4
- MathSciNet review: 1459117