Annular regions containing all the zeros of a polynomial

Suhail Gulzar (), N. A. Rather and K. A. Thakur ()
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Suhail Gulzar: S.P. College
N. A. Rather: University of Kashmir

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 2, 705-710

Abstract: Abstract A result due to Tôya, Montel and Kuniyeda concerning the location of the zeros of a polynomial states that if $$P(z)=a_{n}z^n+a_{n-1}z^{n-1}+a_{n-2}z^{n-2}+\cdots +a_0$$ P ( z ) = a n z n + a n - 1 z n - 1 + a n - 2 z n - 2 + ⋯ + a 0 is a polynomial of degree n then all its zeros lie in the disk $$ |z|\le \left( 1+A_{p}^{q}\right) ^{1/q} $$ | z | ≤ 1 + A p q 1 / q where $$p>1,$$ p > 1 , $$q>1$$ q > 1 with $$1/p+1/q=1$$ 1 / p + 1 / q = 1 and $$A_{p}=\left( \sum _{j=0}^{n-1}\left| \frac{a_j}{a_n}\right| ^p\right) ^{1/p}.$$ A p = ∑ j = 0 n - 1 a j a n p 1 / p . In this paper, we refine this result and among other things obtain ring shaped regions containing all the zeros of a polynomial with complex coefficients.

Keywords: Polynomials; Bounds for zeros; Location of zeros; 30C10; 30C15 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13226-023-00515-x

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