Comptes Rendus
no. 7
Complex analysis
On a family of extremal polynomials
[Sur une famille de polynômes extrémaux]

Présenté par : the Editorial Board

Dmitriy Dmitrishin1  ; Andrey Smorodin1  ; Alex Stokolos2
1 Odessa National Polytechnic University, 1 Shevchenko Ave., Odessa 65044, Ukraine
2 Georgia Southern University, Statesboro, GA 30460, USA
Comptes Rendus. Mathématique, Volume 357 (2019) no. 7, pp. 591-596.

Pour une paire de polynômes trigonométriques C(t)=∑j=1Najcos⁡jt, S(t)=∑j=1Najsin⁡jt à coefficients réels avec la normalisation a1=1, on trouve la valeur extrémale

supa2,…,aN⁡mint⁡{C(t):S(t)=0}=−14sec2⁡πN+2.
Une application en analyse géométrique complexe est montrée. On formule quelques conjectures pour les problèmes extrémaux sur les classes de polynômes.

For a pair of conjugate trigonometrical polynomials C(t)=∑j=1Najcos⁡jt, S(t)=∑j=1Najsin⁡jt with real coefficients and normalization a1=1 the following extremal value is found:

supa2,…,aN⁡mint⁡{C(t):S(t)=0}=−14sec2⁡πN+2.
An application of this result in geometric complex analysis is shown. Several conjectures for a number of extremal problems on classes of polynomials are suggested.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2019.06.010
Dmitriy Dmitrishin 1 ; Andrey Smorodin 1 ; Alex Stokolos 2

1 Odessa National Polytechnic University, 1 Shevchenko Ave., Odessa 65044, Ukraine
2 Georgia Southern University, Statesboro, GA 30460, USA 
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Dmitriy Dmitrishin; Andrey Smorodin; Alex Stokolos. On a family of extremal polynomials. Comptes Rendus. Mathématique, Volume 357 (2019) no. 7, pp. 591-596. doi : 10.1016/j.crma.2019.06.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2019.06.010/

[1] H.P. Boas; D. Khavinson Bohr's power series theorem in several variables, Proc. Amer. Math. Soc., Volume 125 (1997), pp. 2975-2979

[2] H. Bohr A theorem concerning power series, Proc. Lond. Math. Soc. (2), Volume 13 (1914), pp. 1-5

[3] C. Chu Asymptotic Bohr radius for the polynomials in one complex variable, Invariant Subspaces of the Shift Operator, Contemp. Math., vol. 638, Amer. Math. Soc., Providence, RI, USA, 2015, pp. 39-43

[4] D. Dimitrov Extremal positive trigonometric polynomials (B. Bojanov, ed.), Approximation Theory: a Volume Dedicated to Blagovest Sendov, Darba, 2002, pp. 1-24

[5] D. Dmitrishin; K. Dyakonov; A. Stokolos Univalent polynomials and Koebe's one-quarter theorem, Anal. Math. Phys. (2019) | arXiv | DOI

[6] D. Dmitrishin; P. Hagelstein; A. Khamitova; A. Korenovskyi; A. Stokolos Fejér polynomials and control of nonlinear discrete systems, Constr. Approx. (2019) | arXiv | DOI

[7] D. Dmitrishin; P. Hagelstein; A. Khamitova; A. Stokolos Limitations of robust stability of a linear delayed feedback control, SIAM J. Control Optim., Volume 56 (2018), pp. 148-157

[8] D.V. Dmitrishin; A.D. Khamitova Methods of harmonic analysis in nonlinear dynamics, C. R. Acad. Sci. Paris, Ser. I, Volume 351 (2013), pp. 367-370

[9] D.V. Dmitrishin; A. Smorodin; A. Stokolos Estimates of Koebe radius for polynomials, 2018 | arXiv

[10] V.N. Dubinin Methods of geometric function theory in classical and modern problems for polynomials, Russ. Math. Surv., Volume 67 (2012), pp. 599-684

[11] L. Fejér Über trigonometrische Polynome, J. Reine Angew. Math., Volume 146 (1916), pp. 53-82

[12] R. Fournier Asymptotics of the Bohr radius for polynomials of fixed degree, J. Math. Anal. Appl., Volume 338 (2008), pp. 1100-1107

[13] G.M. Goluzin Geometric Theory of Functions of a Complex Variable, Translations of Mathematical Monographs, vol. 26, American Mathematical Society, Providence, RI, USA, 1969

[14] R.E. Greene; S.G. Krantz Function Theory of One Complex Variable, Grad. Stud. Math., vol. 40, American Mathematical Society, Providence, RI, USA, 2006

[15] Z. Guadarrama Bohr's radius for polynomials in one complex variable, Comput. Methods Funct. Theory, Volume 5 (2005), pp. 143-151

[16] D.E. Longsine; S.F. McCormick Simultaneous Rayleigh-quotient minimization methods for Ax=λBx, Linear Algebra Appl., Volume 34 (1980), pp. 195-234

[17] R.S. Martin; J.H. Wilkinson Reduction of the symmetric eigenproblem Ax=λBx and related problems to standard form, Numer. Math., Volume 11 (1968), pp. 99-110

[18] G. Pólya; G. Szegö Problems and Theorems in Analysis. II. Theory of Functions, Zeros, Polynomials, Determinants, Number Theory, Geometry, Springer, Berlin, 1998

[19] W.W. Rogosinski; G. Szegö Extremum problems for non-negative sine polynomials, Acta Sci. Math. Szeged, Volume 12 (1950), pp. 112-124

[20] G. Strang Linear Algebra and Its Applications, Cengage Learning, Boston, MA, USA, 2006

[21] T. Suffridge On univalent polynomials, J. Lond. Math. Soc., Volume 44 (1969), pp. 496-504

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