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arXiv · 2005.12432

On C. Michel's hypothesis about the modulus of typically real polynomials

Dmitriy Dmitrishin·Andrey Smorodin·Alex Stokolos
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Abstract

Extremal problems for typically real polynomials go back to a paper by W. W. Rogosinski and G. Szeg\H{o}, where a number of problems were posed, which were partially solved by using orthogonal polynomials. Since then, not too many new results on extremal properties of typically real polynomials have been obtained. Fundamental work in this direction is due to M.~Brandt, who found a novel way of solving extremal problems. In particular, he solved C. Michel's problem of estimating the modulus of a typically real polynomial of odd degree. On the other hand, D. K. Dimitrov showed the effectivity of Fej\'er's method for solving the Rogosinski--Szeg\H{o} problems. In this article, we completely solve Michel's problem by using Fej\'er's method.

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Dmitriy Dmitrishin, Andrey Smorodin, Alex Stokolos. 2020-05-25. On C. Michel's hypothesis about the modulus of typically real polynomials. https://arxiv.org/abs/2005.12432

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