Geodesic
Estimation of an algebraic polynomial in a plane in terms of its real part on the unit circle
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2010), pp. 92-96 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider the class $\mathcal P_n^*$ of algebraic polynomials of a complex variable with complex coefficients of degree at most $n$ with real constant terms. In this class we estimate the uniform norm of a polynomial $P_n\in\mathcal P_n^*$ on the circle $\Gamma_r=\{z\in\mathbb C\colon|z|=r\}$ of radius $r>1$ in terms of the norm of its real part on the unit circle $\Gamma_1$. More precisely, we study the best constant $\mu(r,n)$ in the inequality $\|P_n\|_{C(\Gamma_r)}\leq\mu(r,n)\|\operatorname{Re}P_n\|_{C(\Gamma_1)}$. We prove that $\mu(r,n)=r^n$ for $r^{n+2}-r^n-3r^2-4r+1\geq0$. In order to justify this result, we obtain the corresponding quadrature formula. We give an example which shows that the strict inequality $\mu(r,n)>r^n$ is valid for $r$ sufficiently close to 1.
Keywords: inequalities for algebraic polynomials, uniform norm, circle in complex plane. 
@article{IVM_2010_3_a11,
     author = {A. V. Parfenenkov},
     title = {Estimation of an algebraic polynomial in a~plane in terms of its real part on the unit circle},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {92--96},
     year = {2010},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2010_3_a11/}
}
TY  - JOUR
AU  - A. V. Parfenenkov
TI  - Estimation of an algebraic polynomial in a plane in terms of its real part on the unit circle
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2010
SP  - 92
EP  - 96
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/IVM_2010_3_a11/
LA  - ru
ID  - IVM_2010_3_a11
ER  - 
%0 Journal Article
%A A. V. Parfenenkov
%T Estimation of an algebraic polynomial in a plane in terms of its real part on the unit circle
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2010
%P 92-96
%N 3
%U http://geodesic.mathdoc.fr/item/IVM_2010_3_a11/
%G ru
%F IVM_2010_3_a11
A. V. Parfenenkov. Estimation of an algebraic polynomial in a plane in terms of its real part on the unit circle. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2010), pp. 92-96. http://geodesic.mathdoc.fr/item/IVM_2010_3_a11/

[1] Szegö G., “Über einen Satz des Hern Serge Bernstein”, Schrift. Königsberg. Gelehrten Gesellschaft., 5:4 (1928), 59–70 | Zbl

[2] Bernshtein S. N., “Ob odnoi teoreme Sege”, Sobr. soch., T. 2, Izd-vo AN SSSR, M., 1954, Statya 62

[3] Frappier C., Rahman Q. I., Ruscheweyh St., “New inequalities for polynomials”, Trans. Amer. Math. Soc., 288 (1985), 69–99 | DOI | MR | Zbl

[4] Dubinin V. N., “Konformnye otobrazheniya i neravenstva dlya algebraicheskikh polinomov”, Algebra i analiz, 13:5 (2001), 16–43 | MR | Zbl

[5] Govil N. K., “On growth of polynomials”, J. Inequal. Appl., 7:5 (2002), 623–631 | DOI | MR | Zbl

[6] Taikov L. V., “Ravnomernaya otsenka velichiny sopryazhennogo polinoma na ploskosti”, Matem. zametki, 54:6 (1993), 142–145 | MR | Zbl

[7] Zigmund A., Trigonometricheskie ryady, T. 1, Mir, M., 1965, 616 pp. | MR