On Critical Values of Polynomials with Real Critical Points |
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Authors: | Aimo Hinkkanen Ilgiz Kayumov |
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Affiliation: | 1. Department of Mathematics, University of Illinois at Urbana–Champaign, 1409 West Green Street, Urbana, IL, 61801, USA 2. Institute of Mathematics and Mechanics, Kazan State University, Universitetskaya 17, 420 008, Kazan, Russia
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Abstract: | Let f be a polynomial of degree at least 2 with f(0)=0 and f′(0)=1. Suppose that all the zeros of f′ are real. We show that there is a zero ζ of f′ such that |f(ζ)/ζ|≤2/3, and that this inequality can be taken to be strict unless f is of the form f(z)=z+cz 3. |
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