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On the location of critical points of polynomials

Published online by Cambridge University Press:  09 April 2009

Abdul Aziz
Abdul Aziz
Affiliation:
Post-graduate Department of Mathematics University of KashmirHazratbal Srinagar-190006Kashmir, India
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Abstract

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Let all the zeros of a polynomial P(z) of degree n lie in |z|≤ 1 and a be a given complex number. In this paper we study the location of the zeros of higher derivatives of the polynomial (z – z) P(z) and obtain certain generalizations of some results of Rahman and Rubinstein. We shall also extend a result of Goodman, Rahman and Ratti for the zeros of the polar derivative of the polynomial P(z) given P(l) = 0.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 36 , Issue 1 , February 1984 , pp. 4 - 11
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Goodman, A. W., Rahman, Q. I. and Ratti, J. S., ‘On the zeros of a poiynomial and its derivative,’ Proc. Amer. Math. Soc. 21 (1969), 273–274.CrossRefGoogle Scholar
[2]Marden, M., ‘Geometry of polynomials’, 2nd ed., Mathematical Surveys 3 (Amer. Math. Soc., Providence, RI., 1966).Google Scholar
[3]Rahman, Q. I., ‘On the zeros of a polynomial and its derivative,’ Pacific J. Math. 41 (1972), 525–528.CrossRefGoogle Scholar
[4]Rubinstein, Z., ‘On a problem of Ilyeff,’ Pacific J. Math. 26 (1968), 158–161.CrossRefGoogle Scholar