Върху някои неравенства за полиноми с реални нули
Ключови думи :
Laguerre–P´olya class, Jensen inrqualities, real–root polynomials, zeros of classical orthogonal polynomialsАбстракт
В статията се разглеждат някои неравенства за полиноми с реални нули, които представляват уточнения на неравенствата на Йенсен за цели функции от класа на Лагер–Пойа. Демонстрира се как такива неравенства могат да се прилагат за получаване на оценки за крайните нули на класическите ортогонални полиноми.
Литература (библиография)
T. Craven, G. Csordas. Jensen polynomials and the Tur´an and Laguerre inequalities. Pacific J. Math. 136, no. 2 (1989), 241–260.
T. Craven, G. Csordas. Iterated Laguerre and Tur´an inequalities. J. Inequal. Pure Appl. Math. 3, No. 3 (2002), article 39, http://jipam.vu.edu.au.
G. Csordas, R. S. Varga. Necessary and sufficient conditions and the Riemann hypothesis. Adv. Appl. Math. 11, no. 3 (1990), 328–352.
D. K. Dimitrov, G. P. Nikolov. Sharp bounds for the extreme zeros of classical orthogonal polynomials. J. Approx. Theory 162, no. 10 (2010), 1793–1804.
W. H. Foster, I. Krasikov. Inequalities for real-root polynomials and entire functions, Adv. Appl. Math. 29, no. 1 (2002), 102–114.
G. H. Hardy, J. E. Littlewood, G. Polya. ´ “Inequalities”, Cambridge University Press, 1934.
J. L. W. V. Jensen. Recherches sur la t´eorie des ´equations, Acta Math. 36, no. 1 (1913), 181–195.
E. Laguerre. Oeuvres. Vol. 1, 2nd ed. Chelsea, New York, 1972.
B. Ja. Levin, Distribution of Zeros of Entire Functions. Translated from the Russian by R. P. Boas, J. M. Danskin, F. M. Goodspeed, J. Korevaar, A. L. Shields and H. P. Thielman. Revised edition. Translations of Mathematical Monographs, vol. 5. Providence, R.I., American Mathematical Society, 1980.
J. B. Love. Problem E 1532. Amer. Math. Monthly 69, no. 7 (1962), 668.
G. Nikolov. Inequalities of Duffin–Schaeffer type, II. East J. Approx. 11, no. 2 (2005), 147–168.
G. Nikolov, R. Uluchev, Inequalities for real-root polynomials. Proof of a conjecture of Foster and Krasikov. In: “Approximation Theory: A Volume Dedicated to Borislav Bojanov” (Eds D. K. Dimitrov, G. Nikolov, R. Uluchev) Sofia, Marin Drinov Academic Publishing House, 2004, 201–216.
N. Obreshkov. On some algebraic covariants and the zeros of polynomials. Izv. Inst. Mat. Acad. Bulgare Sci. 7 (1963), 89–126.
N. Obreshkov. Zeros of Polynomials. Bulgarian Academic Monographs vol. 7. Sofia, Marin Drinov Academic Publishing House, 2003.
M. Patrick. Some inequalities concerning Jacobi polynomials. SIAM J. Math. Anal. 2 (1971), 213–220.
M. Patrick. Extension of inequalities of the Laguerre and Tur´an type. Pacific J. Math. 44 (1973), 675–682.
G. Polya, I. Schur. ´ Uber zwei Arten von Factorenfolgen in der Teorie der algebra ¨ ischen Glaichungen, J. f¨ur Math. 144 (1914), 89–113.
G. Szego. ˝ Orthogonal Polynomials, 4th edn. AMS Colloquium Publications, vol. XXIII, Providence, RI, AMS, 1975.
