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.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
R. Aron, B. Beauzamy and P. Enflo, Polynomials in many variables: real vs. complex norms, J. Approx. Theory, 74 (1993), 181–198.
R. Aron and M. Klimek, Supremum norms for quadratic polynomials, Arch. Math. (Basel), 76 (2001), 73–80.
Contests in Higher Mathematics, Mikl´os Schweitzer Competitions 1962-1991, G. J. Székely, Ed., Problem Books in Mathematics, Springer-Verlag (New York, 1996).
Ph. Glesser, M. Mignotte and M. Petković, Each univariate complex polynomial has a ‘big’ factor, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 71–76.
P. Jiménez-Rodríguez, G. A. Muñoz-Fernández, D. Pellegrino and J. B. Seoane-Sepúlveda, Bernstein-Markov type inequalities and other interesting estimates for polynomials on circle sectors, Math. Inequal. Appl., 20 (2017), 285–300.
S. G. Kim, Exposed 2-homogeneous polynomials on the two-dimensional real predual of Lorentz sequence space, Mediterr. J. Math., 13 (2016), 2827–2839.
S. V. Konyagin and V. F. Lev, On the maximum value of polynomials with given degree and number of roots, Chebyshevskiĭ Sb., 3 (2002), 165–170.
M. Mignotte and Ph. Glesser, On the smallest divisor of a polynomial, J. Symbolic Comput., 17 (1994), 277–282.
G. A. Muñoz-Fernández, D. Pelegrino, J. B. Seoane-Sepúlveda and A. Weber, Supre-mum norms for 2-homogeneous polynomials on circle sectors, J. Convex Anal., 21 (2014), 745–764.
G. A. Muñoz-Fernández and J. B. Seoane-Sepúlveda, Geometry of Banach spaces of trinomials, J. Math. Anal. Appl., 340 (2008), 1069–1087.
S. Neuwirth, The maximum modulus of a trigonometric trinomial, J. Anal. Math., 104 (2008), 371–396.
The maximum of a polynomial on the unit circle, Question 64099 at mathoverflow, https://mathoverflow.net/questions/64099.
C. Visser, A simple proof of certain inequalities concerning polynomials, Nederl. Akad. Wetensch., Proc., 48 (1945), 276–281.
I thank the referee for pointing out the reference .
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.
About this article
Cite this article
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
Key words and phrases
- complex polynomial
- maximum in the unit disc
- inequality between norms
Mathematics Subject Classification