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Zero-free regions for polynomials and some generalizations of Eneström- Kakeya theorem. (English) Zbl 0586.30008

Let P(z)=a n z n +a p z p +···+a 1 z+a 0 [z] satisfy a p a n 0 for some p with 0p<n. By applying the Gershgorin theorem to a matrix which is similar to the companion matrix of P(z), the authors show that for each r>0 all zeros of P(z) lie in |z|max{r, j=0 p |a j /a n |r j+1-n }. From this the authors derive the following main results. Theorem 1. If for some t>0 the above P(z) satisfies |a n |t n-j |a j |,0jp, then all zeros of P(z) lie in |z|K 1 /t, where K 1 (1) is the largest positive root of K n+1 -K n -K p +1=0. Theorem 4. If for some t>0 and some k{0,1,···,n} P(z)= j=0 n a j z j [z],n, satisfies

t n |a n |t n-1 |a n-1 |···t k |a k |t k-1 |a k-1 |···t|a 1 ||a 0 |,

then all zeros of P(z) lie in

|z|t{(2t k |a k |/t n |a n |)-1}+2 j=0 n |a j -|a j ||/|a n |t n-j-1 ·

Next, by using Bernstein’s inequality and a special case of Theorem 1, the authors derive Theorem 2. If P(z)= j=0 n a j z j [z],a>0 are given and if α satisfies max |z|=a |P(z)|=|P(ae iα )|, then P(z)0 holds for |z-ae iα |<a/2n. Finally, several special cases are discussed and generalizations of Theorem 4 are stated.

Reviewer: H.-J.Runckel
MSC:
30C15Zeros of polynomials, etc. (one complex variable)
30C10Polynomials (one complex variable)
30A10Inequalities in the complex domain