Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-26T07:51:00.321Z Has data issue: false hasContentIssue false
  • English
  • Français

Extension of a theorem of J. H. Grace to transcendental entire functions

Published online by Cambridge University Press:  24 October 2008

J. Clunie  and
Q. I. Rahman
J. Clunie
Affiliation:
Department of Mathematics, University of York, Heslington, York, Y01 5DD
Q. I. Rahman
Affiliation:
Département de Mathématiques et de Statistique, Université de Montréal, Montréal, Québec H3C 3J7, Canada
Get access Rights & Permissions [Opens in a new window]

Extract

The following result is due to J. H. Grace (see [4], p. 356, also see [9], §3).

Theorem A. if p is a polynomial such that p(–1) = p(1) then the derivative p' has a zero in each of the half-planes {Rez ≤0} and {Rez ≥0}.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 112 , Issue 3 , November 1992 , pp. 565 - 573
Copyright
Copyright © Cambridge Philosophical Society 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Ahlfors, L. V.. Complex Analysis, 2nd edition (McGraw-Hill Book Company, 1966).Google Scholar
[2]Boas, R. P. Jr. Entire Functions (Academic Press, 1954).Google Scholar
[3]Bojanov, B. D., Rahman, Q. I. and Szynal, J.. On a conjecture of Sendov about the critical points of a polynomial. Math. Z. 190 (1985), 281–285.Google Scholar
[4]Grace, J. H.. The zeros of a polynomial. Proc. Cambridge Philos. Soc. 11 (1902), 352–357.Google Scholar
[5]Heawood, P. J.. Geometrical relations between the roots of f(x) = 0 f′(x) = 0. Quart. J. Pure Appl. Math. 38 (1907), 84–107.Google Scholar
[6]Lindelöf, E.. Sur les fonctions entières d'ordre entier. Ann. Sci. ÉEcole Norm. Sup. (3) 22 (1905), 369–395.CrossRefGoogle Scholar
[7]Marden, M.. Geometry of Polynomials. Mathematical Surveys no. 3 (American Mathematical Society, 1966).Google Scholar
[8]Montel, P.. Leçons sur les Fonctions Univalentes (Gauthier-Villars, 1933).Google Scholar
[9]Szegö, G.. Bemerkungen zu einem Satz von J. H. Grace über die Würzeln algebraischer Gleichungen. Math. Z. 13 (1922), 28–55.CrossRefGoogle Scholar