On a problem of Ilyeff

Rubinstein, Zalman

doi:10.2140/pjm.1968.26.159

Cited by 37 publications

(23 citation statements)

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“…It is not difficult to see that that max p∈S(n,1) |p| 1 = 1 (cf. [15]). On the other hand, Theorem 1 shows that max p∈S(n,0) |p| 0 increases to 1 as n → ∞.…”

Section: Propositionmentioning

confidence: 99%

Maximal and linearly inextensible polynomials

Borcea¹

2006

MATH. SCAND.

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Abstract. Let S(n, 0) be the set of monic complex polynomials of degree n ≥ 2 having all their zeros in the closed unit disk and vanishing at 0. For p ∈ S(n, 0) denote by |p| 0 the distance from the origin to the zero set of p ′ . We determine all 0-maximal polynomials of degree n, that is, all polynomials p ∈ S(n, 0) such that |p| 0 ≥ |q| 0 for any q ∈ S(n, 0). Using a second order variational method we then show that although some of these polynomials are linearly inextensible, they are not locally maximal for Sendov's conjecture.

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“…It is not difficult to see that that max p∈S(n,1) |p| 1 = 1 (cf. [15]). On the other hand, Theorem 1 shows that max p∈S(n,0) |p| 0 increases to 1 as n → ∞.…”

Section: Propositionmentioning

confidence: 99%

Maximal and linearly inextensible polynomials

Borcea¹

2006

MATH. SCAND.

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“…The following corollary is obtained from Theorem 2 by taking r = k. The result (12) is due to Goodman, Rahman and Ratti [1] and (13) was proved by Rubinstein [4].…”

Section: / / All the Zeros Of A Polynomial P(z) Of Degree N Lie In \Zmentioning

confidence: 84%

On the location of critical points of polynomials

Aziz

1984

J Aust Math Soc A

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Let all the zeros of a polynomial P(z) of degree n he 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 J»<1) = 0.

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“…(i) In the special case n = 4, & = 1, the above theorem gives an improvement on Theorem 2 of [5], since it guarantees the existence of a zero of…”

Section: N -K (N -K -L)(n -K)mentioning

confidence: 99%

“…Its validity for polynomials of degree <; 4 was proved in [1] and [5]. Rubinstein has shown in [5] that the statement holds in general if \z ά \ -1.…”

mentioning

confidence: 96%