Generalizations of Gonçalves’ inequality
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- by Peter Borwein, Michael J. Mossinghoff and Jeffrey D. Vaaler PDF
- Proc. Amer. Math. Soc. 135 (2007), 253-261 Request permission
Abstract:
Let $F(z)=\sum _{n=0}^N a_n z^n$ be a polynomial with complex coefficients and roots $\alpha _1$, …, $\alpha _N$, let $\|F\|_p$ denote its $L_p$ norm over the unit circle, and let $\|F\|_p$ denote Mahler’s measure of $F$. Gonçalves’ inequality asserts that \begin{align*} \|F\|_2 &\geq |a_N| \left ( \prod _{n=1}^N \max \{1, |\alpha _n|^2\} + \prod _{n=1}^N \min \{1, |\alpha _n|^2\} \right )^{1/2} &= \|F\|_0\left (1+\frac {|a_0 a_N|^2}{\|F\|^4}\right )^{1/2}. \end{align*} We prove that \[ \|F\|_p \geq B_p |a_N| \left ( \prod _{n=1}^N \max \{1, |\alpha _n|^p\} + \prod _{n=1}^N \min \{1, |\alpha _n|^p\} \right )^{1/p} \] for $1\leq p\leq 2$, where $B_p$ is an explicit constant, and that \[ \|F\|_p \geq \|F\|_0 \left (1+\frac {p^2|a_0 a_N|^2}{4\|F\|^4}\right )^{1/p} \] for $p\geq 1$. We also establish additional lower bounds on the $L_p$ norms of a polynomial in terms of its coefficients.References
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Additional Information
- Peter Borwein
- Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
- Michael J. Mossinghoff
- Affiliation: Department of Mathematics, Davidson College, Davidson, North Carolina 28035
- MR Author ID: 630072
- ORCID: 0000-0002-7983-5427
- Email: mjm@member.ams.org
- Jeffrey D. Vaaler
- Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712
- MR Author ID: 176405
- Email: vaaler@math.utexas.edu
- Received by editor(s): January 6, 2005
- Received by editor(s) in revised form: August 8, 2005
- Published electronically: June 30, 2006
- Additional Notes: The research of P. Borwein was supported in part by NSERC of Canada and MITACS
- Communicated by: Juha M. Heinonen
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 135 (2007), 253-261
- MSC (2000): Primary 30A10, 30C10; Secondary 26D05, 42A05
- DOI: https://doi.org/10.1090/S0002-9939-06-08454-1
- MathSciNet review: 2280202