Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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ON SENDOV'S CONJECTURE ABOUT CRITICAL POINTS OF A POLYNOMIAL

  • Received : 2021.09.16
  • Accepted : 2021.12.10
  • Published : 2021.12.30
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Abstract

The derivative of a polynomial p(z) of degree n, with respect to point α is defined by Dαp(z) = np(z) + (α - z)p'(z). Let p(z) be a polynomial having all its zeros in the unit disk |z| ≤ 1. The Sendov conjecture asserts that if all the zeros of a polynomial p(z) lie in the closed unit disk, then there must be a zero of p'(z) within unit distance of each zero. In this paper, we obtain certain results concerning the location of the zeros of Dαp(z) with respect to a specific zero of p(z) and a stronger result than Sendov conjecture is obtained. Further, a result is obtained for zeros of higher derivatives of polynomials having multiple roots.

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References

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