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The zeros of a certain family of trinomials

Published online by Cambridge University Press:  18 May 2009

Karl Dilcher ,
James D. Nulton  and
Kenneth B. Stolarsky
Karl Dilcher
Affiliation:
Dept. of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, CanadaB3H 3J5
James D. Nulton
Affiliation:
2045 Montclair Street, San Diego, California 92104, U.S.A.
Kenneth B. Stolarsky
Affiliation:
Dept. of Mathematics, 1409 West Green, University of Illinois, Urbana, Illinois 61801, U.S.A.
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Many of the classical inequalities of analysis can be written in the form P(x) ≥ 0 for x ∈ I or P(x) > 0 for x ∈ I′, where P(x) is a polynomial and I′ ⊂ I are certain intervals on the real line. This gives rise to the question of where the zeros of P(x) are located. For example, if f is a polynomial with real zeros, then an inequality of Laguerre [8, p. 171 f.] asserts that

for all x. A detailed study of the zeros of this particular P(x) has been made [5].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 34 , Issue 1 , January 1992 , pp. 55 - 74
Copyright
Copyright © Glasgow Mathematical Journal Trust 1992

References

REFERENCES

1.Beckenbach, E. F. and Bellman, R. E., Inequalities (Springer, Berlin, 1961).CrossRefGoogle Scholar
2.Bellman, R. E. and Cooke, K. L., Differential-difference equations (Academic Press, New York 1963).Google Scholar
3.Boese, G., Einschliisse und Trennung der Nullstellen von Exponentialtrinomen, Z. Angew. Math. Mech. 62 (1982), 547–560.CrossRefGoogle Scholar
4.Dilcher, K., A generalization of the Enestrom-Kakeya theorem, J. Math. Anal. Appl. 116 (1986), 473–488.CrossRefGoogle Scholar
5.Dilcher, K. and Stolarsky, K. B., Zeros of the Wronskian of a polynomial, J. Math. Anal. Appl. (to appear).Google Scholar
6.Evans, R. J. and Stolarsky, K. B., A family of polynomials with concyclic zeros II, Proc. Amer. Math. Soc. 92 (1984) 393–396.CrossRefGoogle Scholar
7.Hardy, G. H., Littlewood, J. E., and Pólya, G., Inequalities (Cambridge Univ. Press, 1964).Google Scholar
8.Laguerre, E., Oeuvres, Vol. 1, 2nd ed. (Chelsea Publishing, New York, 1972).Google Scholar
9.Marden, M., Geometry of polynomials, Mathematical Surveys (3), (Amer. Math. Soc., Providence 1966).Google Scholar
10.Nicolas, J. L. and Schinzel, A., Localisation des zeros de polynõmes intervenant en theorie du signal. In: Langevin, M. and Waldschmidt, M. (Eds.), Cinquante Ans de Polynomes. Lecture Notes in Mathematics 1415, Springer-Verlag 1990.Google Scholar
11.Nulton, J. D. and Stolarsky, K. B., Zeros of certain trinomials, C. R. Math. Rep. Acad. Sci. Canada 6 (1984), 243–248.Google Scholar
12.Pólya, G. and Szegö, G., Problems and Theorems in Analysis, 1, 4th ed., (Springer Verlag, Berlin 1970).Google Scholar
13.Rahman, Q. I. and Szynal, J., On some classes of polynomials, Canad. J. Math. 30 (1978), 332–349.CrossRefGoogle Scholar
14.Rahman, Q. I. and Waniurski, J., Coefficient regions for univalent trinomials, Canad. J. Math. 32 (1980), 1–20.CrossRefGoogle Scholar
15.Stolarsky, K. B., Zeros of exponential polynomials and “reductionism”, Topics in Classical Number Theory, Coll. Math. Soc. Jdnos Bolyai 34, Elsevier North Holland, Amsterdam 1984.Google Scholar
16.Stolarsky, K. B., A family of polynomials with concyclic zeros, Proc. Amer. Math. Soc. 88 (1983), 622–624.CrossRefGoogle Scholar
17.Stolarsky, K. B., A family of polynomials with concyclic zeros III, Quart. J. Math. Oxford (2) 36 (1985), 255–259.CrossRefGoogle Scholar
18.Yates, R. C., A Handbook on Curves and their Properties (Edwards, J. W., Ann Arbor 1959).Google Scholar