On a theorem of J. L. Walsh concerning the moduli of zeros of polynomials

Summary

Alexander Ostrowski recently gave a proof of a result which is a conditioned converse of a theorem of A. E. Pellet. The theorem proved by Ostrowski is due to J. L. Walsh, but it was pointed out by Ostrowski that Walsh’s proof of his theorem was incomplete. Ostrowski’s proof utilizes the theorem of Rouché whereas the result proved concerns the moduli of zeros of polynomials. This note contains a proof of Walsh’s result by elementary reasoning which is more appropriate to the subject. The result proved is thatif P(z)=a ₀+a₁z+...a_nzⁿ, and α is such that for z=α the modulus of no term of P(z) exceeds the sum of the moduli of all the other terms of P(z) then α is a root of some equation b₀+b₁z+...+b_nzⁿ=0 where |a_r|=|b_r|, (r=0, 1, 2, ..., n).