Location of the zeros of polynomials with a prescribed norm

Rahman, Q. I.; Schmeisser, G.

doi:10.1090/S0002-9947-1974-0349968-1

Location of the zeros of polynomials with a prescribed norm
HTML articles powered by AMS MathViewer

by Q. I. Rahman and G. Schmeisser PDF

Trans. Amer. Math. Soc. 196 (1974), 69-78 Request permission

Abstract:

For monic polynomials ${f_n}(z)$ of degree n with prescribed ${L^p}$ norm $(1 \leq p \leq \infty )$ on the unit circle or supremum norm on the unit interval we determine bounded regions in the complex plane containing at least $k(1 \leq k \leq n)$ zeros. We deduce our results from some new inequalities which are similar to an inequality of Vicente Gonçalves and relate the zeros of a polynomial to its norm.

References

L’approximation

40

André Giroux, Estimates for the imaginary parts of the zeros of a polynomial, Proc. Amer. Math. Soc. 44 (1974), 61–67. MR 333566, DOI 10.1090/S0002-9939-1974-0333566-5
Morris Marden, Geometry of polynomials, 2nd ed., Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972
A. M. Ostrowski, On an inequality of J. Vicente Gonçalves, Univ. Lisboa Rev. Fac. Ci. A (2) 8 (1960), 115–119. MR 145049
Q. I. Rahman, Inequalities concerning polynomials and trigonometric polynomials, J. Math. Anal. Appl. 6 (1963), 303–324. MR 152796, DOI 10.1016/0022-247X(63)90011-8
Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR 0071727
W. Specht, Algebraische Gleichungen mit reelen oder komplexen Koeffizienten, Enzyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, Bd. I, vol. 1, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1958 (German). MR 0106274
E. C. Titchmarsh, Han-shu lun, Science Press, Peking, 1964 (Chinese). Translated from the English by Wu Chin. MR 0197687
J. Vicente Gonçalves, L’inégalité de W. Specht, Univ. Lisboa Rev. Fac. Ci. A (2) 1 (1950), 167–171 (French). MR 39835
C. Visser, A simple proof of certain inequalities concerning polynomials, Nederl. Akad. Wetensch., Proc. 48 (1945), 276–281 = Indagationes Math. 7, 81–86 (1945). MR 15568
A. Zygmund, Two notes on inequalities, J. Math. Phys. Mass. Inst. Tech. 21 (1942), 117–123. MR 7425, DOI 10.1002/sapm1942211117

Trigonometric series

21