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The zeros of linear combinations of translates of polynomials

Part of: Geometric function theory Polynomials, rational functions

Published online by Cambridge University Press:  09 April 2009

Peter Walker
Peter Walker
Affiliation:
College of Arts and Science, American University of Sharjah, P.O. 26666 Sharjah, United Arab Emirates e-mail: [email protected]
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Abstract

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We investigate the location and separation of zeros of certain three-term linear combination of translates of polynomials. In particular, we find an interval of the form I = [−1, 1 + h], h > 0 such that for a polynomial f, all of whose zeros are real, and λ ∈ I, all zeros of f (x + 2ic) + 2λf (x) + f (x – 2ic) are also real.

MSC classification

Secondary: 26C10: Polynomials 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 72 , Issue 1 , February 2002 , pp. 109 - 118
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Pólya, G., ‘Algebraische Untersuchungen über ganze Funktionen vom Geschlechte Null und Eins’, J. Reine Angew. Math. 145 (1915), 224–249.CrossRefGoogle Scholar
[2]Pólya, G., ‘Bemerkung über die Integraldarstellung der Riemannsche -Function’, Acta Math. 48 (1926), 305–317.CrossRefGoogle Scholar
[3]Walker, P. L., ‘Separation of the zeros of polynomials’, Amer. Math. Monthly 100 (1993), 272–273.Google Scholar
[4]Walker, P. L., ‘Bounds for the separation of real zeros of polynomials’, J. Austral. Math. Soc. (Series A) 59 (1995), 330–342.CrossRefGoogle Scholar
[5]Walker, P. L., ‘Separation of zeros of translates of polynomials and entire functions’, J. Math. Anal. Appl. 206 (1997), 270–279.CrossRefGoogle Scholar