On the zeros of polynomials with concentration at low degrees. II (Q1260816)

[For part I see the second author in ibid. 149, No. 2, 424-436 (1990; Zbl 0714.30008).] A complex polynomial \(P(z)=\sum^ n_{j=0} a_ jz^ j\) is defined to have concentration \(d(0<d \leq 1)\) at degree \(k\), if \(\sum^ k_{j=0} | a_ j | \geq d \sum^ n_{j=0} | a_ j |\) holds. Let \(z_ 1,\dots,z_ n\) be the zeros of \(P(z)\) with \(| z_ 1 | \leq | z_ 2 | \leq \cdots \leq | z_ n |\). The main result of the authors is the following theorem. If \(P\) has concentration \(d\) at degree \(k\), then \[ \left | \sum_{j>k}z^{-1}_ j \right | \leq C(d,k):=(9^{k+3}/d^ 3) (2(1+d)/d)^ k \text{ and } \prod_{j>k} | z_ j | \geq 1/C(d,k) \text{ hold}. \] The proof is incomplete, since in its final step the inequality \(C(d/2,k-1) \leq C(d,k)\) is used, which does not hold for large \(k\) and fixed \(d\). Among further results a circle, centered at the origin, is constructed, that contains at least one zero of \(P\) and the following theorem is proved. If \(P\) is a Hurwitz polynomial (i.e. all \(a_ j>0\) and all \(\text{Re} z_ j \leq 0)\) with concentration \(d\) at degree \(k\), then \[ \sum^ n_{j=1} (1-z_ j)^{-1} \leq C_ H(d,k):=9 \log(1/d)+(11k+9) \log 2 \] holds, which also yields an upper bound for \(| \sum_{j>k} z_ j^{-1} |\) and a lower bound for \(\prod_{j>k} | z_ j |\). Furthermore the generalized Bernstein inequality \(\| P' \|\leq C_ H (d,k) \| P \|\) is obtained, where \(\| P \|:=\max_{| z |=1} | P(z) |\). The results on Hurwitz polynomials are extended to special classes of entire functions of order \(\leq 1\).