Vol. 41, No. 2, 1972

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On the zeros of a polynomial and its derivative

Qazi Ibadur Rahman

Vol. 41 (1972), No. 2, 525–528
Abstract

Let all the zeros of a polynomial p(z) of degree n lie in |z|1. Given a complex number a what is the radius of the smallest disk centred at α containing at least one zero of the polynomial ((z a)p(z))f? According to Theorem 1 the answer is (|a| + 1)(n + 1) if |α| > (n + 2)∕n. Theorem 2 which states that if both the zeros of the quadratic polynomial p(z) lie in |z|1 and |a|2 then ((z α)p(z))has at least one zero in

|z − a| ≦ {3|a|+ (12− 3|a|2)1∕2}∕6

completely settles the case n = 2.

Mathematical Subject Classification
Primary: 30A08
Milestones
Received: 28 January 1971
Published: 1 May 1972
Authors
Qazi Ibadur Rahman