Nikiforova, Tatiana M. Inequalities for algebraic polynomials on an ellipse. (English) Zbl 1462.30012 Ural Math. J. 6, No. 2, 87-94 (2020). Summary: The paper presents new solutions to two classical problems of approximation theory. The first problem is to find the polynomial that deviates least from zero on an ellipse. The second one is to find the exact upper bound of the uniform norm on an ellipse with foci \(\pm 1\) of the derivative of an algebraic polynomial with real coefficients normalized on the segment \([- 1,1]\). MSC: 30C10 Polynomials and rational functions of one complex variable 30E10 Approximation in the complex plane Keywords:polynomials; approximation × Cite Format Result Cite Review PDF Full Text: DOI MNR References: [1] Smirnov V. I., Lebedev N. A., Konstruktivnaya teoriya funkcij kompleksnogo peremennogo [The Constructive Theory of Functions of a Complex Variable], Nauka Publ., Leningrad, 1964, 438 pp. (in Russian) · Zbl 0122.31502 [2] Kolmogorov A. N., “A remark on the polynomials of P.L. Chebyshev deviating the least from a given function”, Uspehi Mat. Nauk, 3:1 (1948), 216-221 (in Russian) · Zbl 0030.02803 [3] Kemperman J. H. B., “Markov type inequalities for the derivatives of a polynomial”, Aspects of Mathematics and its Applications, 34 (1986), 465-476 · Zbl 0593.30008 · doi:10.1016/S0924-6509(09)70275-2 [4] Duffin R., Schaeffer A. C., “Some properties of functions of exponential type”, Bull. Amer. Math. Soc., 4:4 (1938), 236-240 · JFM 64.0298.01 · doi:10.1090/S0002-9904-1938-06725-0 [5] Erdös, “Some remark on polynomials”, Bull. Amer. Math. Soc., 53:12 (1947), 1169-1176 · Zbl 0032.38604 · doi:10.1090/S0002-9904-1947-08938-2 [6] Abramowitz M., Stegun I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, NY, 1965, 1046 pp. · Zbl 0171.38503 · doi:10.1119/1.15378 [7] Bernstein S. N., “O nailuchshem priblizhenii nepreryvnykh funktsii posredstvom mnogochlenov dannoi stepeni [On the Best Approximation of Continuous Functions by Polynomials of a Given Degree]”, Comm. Soc. Math. Kharkov. 2 Series, XIII (13):2-5 (1912), 49-194 (in Russian) · JFM 43.0493.01 [8] Dzyadyk V. K., Vvedenie v teoriyu ravnomernogo priblizheniya funkcij polinomami [Introduction to the Theory of Uniform Approximation of Functions by Polynomials], Nauka, Moscow, 1977, 508 pp. (in Russian) · Zbl 0481.41001 [9] Markov A. A., “Ob odnom voproce D.I. Mendeleeva [On a Question by D.I. Mendeleev]”, Zap. Imp. Akad. Nauk., St. Petersburg, 62 (1890), 1-24 (in Russian) [10] Schaeffer A. C., Szegö G., “Inequalities for harmonic polynomials in two and three dimensions”, Trans. Amer. Math. Soc., 50 (1941), 187-225 · JFM 67.1001.03 · doi:10.1090/S0002-9947-1941-0005164-7 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.