Vol. 15, No. 4, 1965

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ISSN: 0030-8730
Some results in the location of zeros of polynomials

Zalman Rubinstein

Vol. 15 (1965), No. 4, 1391–1395
Abstract

Three out of the four theorems proved in this paper deal with the location of the zeros of a polynomial P(z) whose zeros zi, i = 1,2,,n satisfy the conditions |zi|1, and i=1nzip = 0 for p = 1,2,,l. One of those estimates is

 P′′(z)  P-′(z)  1    ----l+-1----
|P′(z) − P (z) − z | < |z|(|z|l+1 − 1)

for |z| > 1.

The fourth result is of a different nature. It refines, in particular, a theorem due to Eneström and Kakeya. It is shown that no zero of the polynomial h(z) = k=0nbkzk lies in the disk

       −i𝜃
|z −-βe---| < --1-,
(β + 1)   β + 1

where β = max|z|=1|h(z)|max|z|=1|h(z)|, and max|z|=1|h(z)| = |h(ei𝜃)|.

Mathematical Subject Classification
Primary: 30.11
Milestones
Received: 3 June 1964
Published: 1 December 1965
Authors
Zalman Rubinstein