Bounds for the zeros of polynomials (Q5916476)

Let \(f(z)=z^ n+a_{n-1}z^{n-1}+...+a_ 1z+a_ 0\) be a polynomial with complex coefficients. Let z be any zero of f. \textit{E. Deutsch} [Am. Math. Monthly 88, 205-206 (1981; Zbl 0458.30007)] gave two upper bounds \(R_ 1\) and \(R_ 3\) for \(| z|\) which generalized classical results of Cauchy. For example, his second bound was \[ | z| \leq \max \{1,| a_ 0| +,...+| a_ k|,1+| a_{k+1}|,...,1+| a_{n-1}| \}=R_ 3, \] for any \(k\in \{0,1,...,n-1\}\). The authors denote these two results as Theorem A and Theorem B. The authors give rather complicated lower bounds \(R_ 2\) and \(R_ 4\), corresponding to \(R_ 1\) and \(R_ 3\), producing annular regions \(R_ 2\leq | z| \leq R_ 1\) (Theorem 1) and \(R_ 4\leq | z| \leq R_ 3\) (Theorem 2) containing all the zeros of f. They claim that, in case \(k=0\) or \(k=n-1\), these give improvements of Cauchy's results while Deutsch's results do not. However, as is well known, one can easily obtain a lower bound for the zeros of a polynomial by applying any upper bound to the reciprocal polynomial. Taking the example \(f(z)=z^ 3+0.3z^ 2+0.7z+0.7\), used by Deutsch and the authors, Deutsch's result quoted above gives \(7/11=.6363...\leq | z| \leq 1.4\) (the lower bound is just Cauchy's estimate). The authors claim a lower bound of \(R_ 4=.67\) for their Theorem 2, with \(R_ 3=1.4\), but it seems that the correct value is \(R_ 4=.32889....\) An inspection of the proof of Theorem 1 shows that no use is made of the fact that \(R_ 1\) is an upper bound of the zeros of f and indeed the proof is for any \(R\geq 1\), provided \(R_ 2\) is taken as the minimum of 1 and the quantity given in their theorem. Taking \(R=1\) gives the lower bound.38592..., which is better than that obtained with \(R=R_ 3=1.4\) but not as good as that obtainable from Cauchy's estimate.