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Generalizations of Gonçalves' inequality

Authors: Peter Borwein, Michael J. Mossinghoff and Jeffrey D. Vaaler
Journal: Proc. Amer. Math. Soc. 135 (2007), 253-261
MSC (2000): Primary 30A10, 30C10; Secondary 26D05, 42A05
Published electronically: June 30, 2006
MathSciNet review: 2280202
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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 $ \Vert F\Vert _p$ denote its $ L_p$ norm over the unit circle, and let $ \Vert F\Vert _p$ denote Mahler's measure of $ F$. Gonçalves' inequality asserts that

$\displaystyle \Vert F\Vert _2$ $\displaystyle \geq \vert a_N\vert \left( \prod_{n=1}^N \max\{1, \vert\alpha_n\vert^2\} + \prod_{n=1}^N \min\{1, \vert\alpha_n\vert^2\} \right)^{1/2}$    
  $\displaystyle = \Vert F\Vert _0\left(1+\frac{\vert a_0 a_N\vert^2}{\Vert F\Vert^4}\right)^{1/2}.$    

We prove that

$\displaystyle \Vert F\Vert _p \geq B_p \vert a_N\vert \left( \prod_{n=1}^N \m... ...lpha_n\vert^p\} + \prod_{n=1}^N \min\{1, \vert\alpha_n\vert^p\} \right)^{1/p} $

for $ 1\leq p\leq2$, where $ B_p$ is an explicit constant, and that

$\displaystyle \Vert F\Vert _p \geq \Vert F\Vert _0 \left(1+\frac{p^2\vert a_0 a_N\vert^2}{4\Vert F\Vert^4}\right)^{1/p} $

for $ p\geq1$. We also establish additional lower bounds on the $ L_p$ norms of a polynomial in terms of its coefficients.

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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

Jeffrey D. Vaaler
Affiliation: Department of Mathematics, University of Texas, Austin, Texas 78712

Keywords: $L_p$ norm, polynomial, Gon\c{c}alves' inequality, Hausdorff-Young inequality
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
Article copyright: © Copyright 2006 American Mathematical Society