Bernstein's asymptotic best bound for the Kth derivative of a polynomial
โ Scribed by Robert Whitley
- Publisher
- Elsevier Science
- Year
- 1985
- Tongue
- English
- Weight
- 462 KB
- Volume
- 105
- Category
- Article
- ISSN
- 0022-247X
No coin nor oath required. For personal study only.
๐ SIMILAR VOLUMES
Let p n z be a polynomial of degree n and D ฮฑ p n z its polar derivative. It has been proved that if p n z has no zeros in z < 1, then for ฮด โฅ 1 and ฮฑ โฅ 1, 2ฯ 0 D ฮฑ p n e iฮธ ฮด dฮธ 1/ฮด โค n ฮฑ + 1 F ฮด 2ฯ 0 p n e iฮธ ฮด dฮธ 1/ฮด where F ฮด = 2ฯ/ 2ฯ 0 1 + e iฮธ ฮด dฮธ 1/ฮด . We also obtain analogous inequalities
Let \(f(x)\) be a polynomial of degree \(n\) with complex coefficients, which factors as \(f(x)=\) \(g(x) h(x)\). Let \(H(g)\) be the maximum of the absolute value of the coefficients of \(g\). For \(1 \leq p \leq \infty\), let \([f]_{p}\) denote the \(p^{\text {th }}\) Bombieri norm of \(f\). This