Abstract
For a polynomial p(z )o f degreen, having all zeros in |z |≤ k ,w herek ≤ 1, Dewan et al. (Southeast Asian Bull. Math. 34:69-77, 2010) proved that for every α ∈ C with |α |≥ k and for each r >0 ,
Highlights
Introduction and statement of resultsLet p(z) be a polynomial of degree n
According to Bernstein’s inequality [ ] on the derivative of a polynomial, we have max p (z) ≤ n max p(z). This result is best possible and equality holds for a polynomial that has all zeros at the origin
If we restrict to the class of polynomials which have all zeros in |z| ≤, it has been proved by Turán [ ] that max p (z)
Summary
) by |α| and make |α| → ∞, we obtain the following refinement and generalization of the inequality ). all its zeros in |z| ≤ k ≤ and m = min|z|=k |p(z)|, for every λ ∈ C with |λ| ≤ and r > , n r max p (z) , Letting r → ∞ in If k = , p(z) has all its zeros at the origin, p(z) = anzn In this case m = , sμ = and DαP(z) = nαanzn– , on the left-hand side of Since all the zeros of p(z) lie in |z| ≤ k, it follows by Rouche’s theorem that all the zeros of.
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