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

TL;DR

This paper completely solves C. Michel's problem on estimating the modulus of typically real polynomials of odd degree by applying Fejér's method, advancing the understanding of extremal properties in this class.

Contribution

The paper introduces a complete solution to Michel's extremal problem using Fejér's method, filling a longstanding gap in the theory of typically real polynomials.

Findings

Michel's problem is fully solved.

Fejér's method is effective for extremal problems.

Advances in understanding modulus bounds for typically real polynomials.

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.