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<p>For a complex polynomial of degree at least two that fixes the origin and has all its critical points on the positive or negative real half-axis, we establish an exact lower bound for the largest modulus of its critical values.This estimate involves the Schwarzian derivative of the polynomial at&#xa0;the origin and is independent of the degree of the polynomial.A similar estimate is given for the case when all the critical points of the polynomial are real and located on opposite sides of the origin.</p>

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The Schwarzian Derivative and Critical Values of a Polynomial with Real Critical Points

  • 期刊论文

  • 发表日期：

    2025年11月24日

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  • V. N. Dubinin

摘要

For a complex polynomial of degree at least two that fixes the origin and has all its critical points on the positive or negative real half-axis, we establish an exact lower bound for the largest modulus of its critical values.This estimate involves the Schwarzian derivative of the polynomial at the origin and is independent of the degree of the polynomial.A similar estimate is given for the case when all the critical points of the polynomial are real and located on opposite sides of the origin.