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A Lower Bound for the Maximum of a Polynomial in the Unit Disc

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Abstract

In this paper we show that the maximum of a polynomial P ∈ ℂ[z] of degree d ≥ 2 in the unit disc can be bounded below by the sum of moduli of its two coefficients, say, for zs and zt under certain assumption on the pair s < t. We also show that this assumption on s, t cannot be removed or weakened and give several examples showing when this lower bound is (or is not) attained.

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Acknowledgement

I thank the referee for pointing out the reference [13].

Author information

Correspondence to A. Dubickas.

Additional information

This research was supported by the European Social Fund according to the activity ‘Improvement of researchers qualification by implementing world-class R&D projects of Measure No. 09.3.3-LMT-K-712-01-0037.

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Dubickas, A. A Lower Bound for the Maximum of a Polynomial in the Unit Disc. Anal Math 46, 67–76 (2020). https://doi.org/10.1007/s10476-020-0017-y

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Key words and phrases

  • complex polynomial
  • maximum in the unit disc
  • inequality between norms

Mathematics Subject Classification

  • 12D99
  • 12E05