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Bohr’s Radius for Polynomials in One Complex Variable

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Abstract

We consider complex polynomials of degree n that are bounded by one in the unit disc and give estimates on the size of the radius R n of the disc where the sum of the moduli of the individual terms of the polynomial is less than one. We find that there are positive constants C 1, C 2 such that

$$C_1 \frac{1} {{3^{{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-\nulldelimiterspace} 2}} }} < R_n - \frac{1} {3} < C_2 \frac{{\log n}} {n}.$$

This result generalizes the celebrated theorem of Harald Bohr to polynomials of degree n.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas, 72701, USA

    Zdeňka Guadarrama

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  1. Zdeňka Guadarrama
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Correspondence to Zdeňka Guadarrama.

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This work was supported in part by the NSF grant DMS-0139008 (PI-D.Khavinson).

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Guadarrama, Z. Bohr’s Radius for Polynomials in One Complex Variable. Comput. Methods Funct. Theory 5, 143–151 (2005). https://doi.org/10.1007/BF03321091

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  • DOI: https://doi.org/10.1007/BF03321091

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