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- W3178993375 abstract "Abstract In this paper, for polynomials with real coefficients <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>P</m:mi> <m:mo>,</m:mo> <m:mi>Q</m:mi> </m:math> P,Q satisfying <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mo>∣</m:mo> <m:mi>P</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∣</m:mo> <m:mo>≤</m:mo> <m:mo>∣</m:mo> <m:mi>Q</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∣</m:mo> </m:math> | Pleft(x)| le | Qleft(x)| for each <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>x</m:mi> </m:math> x in a real interval <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>I</m:mi> </m:math> I , we prove the bound <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>L</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>P</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>≤</m:mo> <m:mi>c</m:mi> <m:mi>L</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>Q</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> Lleft(P)le cLleft(Q) between the lengths of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>P</m:mi> </m:math> P and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>Q</m:mi> </m:math> Q with a constant <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>c</m:mi> </m:math> c , which is exponential in the degree <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>d</m:mi> </m:math> d of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>P</m:mi> </m:math> P . An example showing that the constant <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>c</m:mi> </m:math> c in this bound should be at least exponential in <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>d</m:mi> </m:math> d is also given. Similar inequalities are obtained for the heights of <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>P</m:mi> </m:math> P and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>Q</m:mi> </m:math> Q when the interval <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>I</m:mi> </m:math> I is infinite and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>P</m:mi> <m:mo>,</m:mo> <m:mi>Q</m:mi> </m:math> P,Q are both of degree <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mi>d</m:mi> </m:math> d . In the proofs and in the constructions of examples, we use some translations of Chebyshev polynomials." @default.
- W3178993375 created "2021-07-19" @default.
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- W3178993375 date "2021-01-01" @default.
- W3178993375 modified "2023-09-27" @default.
- W3178993375 title "Inequalities between height and deviation of polynomials" @default.
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- W3178993375 doi "https://doi.org/10.1515/math-2021-0055" @default.
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