On some inequalities for derivatives of algebraic polynomials in unbounded regions with angles

Year 2021, Volume: 9 Issue: 1, 93 - 103, 30.06.2021

Cevahir Doğanay Gün

https://doi.org/10.51354/mjen.846484

Abstract

In this work we study Bernstein-Walsh-type estimations for the derivative of an arbitrary algebraic polynomial in regions with interior zero and exterior non zero angles.

Keywords

Algebraic polynomial, quasicircle, smooth curve, inequalities.

References

• Abdullayev F. G., Andrievskii V. V., “On the orthogonal polynomials in the domains with K-quasiconformal boundary”, Izv. Akad. Nauk Azerb. SSR., Ser. FTM, 1983, 1, 3-7.
• Abdullayev F. G., “On the interference of the weight boundary contour for orthogonal polynomials over the region”, J. of Comp. Anal. and Appl., 6 (1), 31-42, (2004).
• Abdullayev F. G., Özkartepe P., “On the behavior of the algebraic polynomial in unbounded regions with piecewise dini-smooth boundary”, Ukr. Math. J. , Vol. 66 , No: 5, 2014, pp. 645-665.
• Abdullayev F. G., Gün C. D., “On the behavior of the algebraic polynomials in regions with piecewise smooth boundary without cusps”, Ann. Polon. Math., 2014, 111, 39-58.
• Abdullayev F. G., Gün C. D., Özkartepe P., “Inequalities for algebraic polynomials in regions with exterior cusps”, J. Nonlinear Funct. Anal. Article ID 3, 1-32, (2015).
• Abdullayev F. G., Özkartepe P., “On the growth of algebraic polynomials in the whole complex plane”, J. Korean Math. Soc. 52(4, 699-725, (2015).
• Abdullayev F. G., Özkartepe P., “Uniform and pointwise polynomial inequalities in regions with cusps in the weighted Lebesgue space”, Jaen Journal on Approximation 7(2), 231-261, (2015).
• Abdullayev F. G., Özkartepe P., “Polynomial inequalities in Lavrentiev regions with interior and exterior zero angles in the weighted Lebesgue space”, Publications de l’Institut Mathématique (Beograd) 100 (114), 209-227, (2016).
• Abdullayev F. G., Aral D., “On the Bernstein-Walsh type Lemmas in regions of the complex plane”, Ukr. Math. J., Vol. 63 (3), 337-350, (2011).
• Abdullayev F. G., Özkartepe P., “An analogue of the Bernstein-Walsh lemma in Jordan regions of the complex plane”, Journal Ineq. and Appl., 2013:570 7p, (2013).
• Abdullayev F. G., Özkartepe P., “On the behavior of the algebraic polynomials in unbounded regions with piecewise dini-smooth boundary”, Ukr. Mat. J. – 2014. – 66 (5) – P. 575 – 597.
• Abdullayev F. G., Özkartepe N.P., “Uniform and pointwise Bernstein-Walsh-type inequalities on a quasidisk in the complex plane”, Bull. Belg. Math. Soc., 2016, 23 (2), 285–310.
• Abdullayev F. G., Gün C. D., “Bernstein-Walsh -type inequalities for derivatives of algebraic polynomials”, 2020. (to appear)
• Ahlfors L., “Lectures on quasiconformal mappings”, Princeton, NJ: Van Nostrand, 1966.
• Andrievskii V. V., “Weighted polynomial inequalities in the complex plane”, J. Approx.Theory, 2012, 164 (9), 1165-1183.
• Andrievskii V. V., Belyi V. I., Dzyadyk V. K., “Conformal invariants in cocstructive theory of functions of complex plane”, Atlanta, World Federatin Publ. Com., 1995.
• Belinskii P. P., “General properties of quasiconformal mappings”, Nauka, Sib. otd., Novosibirsk, 1974. [in Russian]
• Dzyadyk V. K., “Introduction to the theory of uniform approximation of function by polynomials”, Nauka, Moskow, 1977.
• Gaier D., “On the convergence of the Bieberbach polynomials in regions with corners”, Constructive Approximation, 4 (1988), pp.289-305.
• Hille E., Szegö G., Tamarkin J. D., “On some generalization of a theorem of A.Markoff”, Duke Math., 3, 729-739, (1937).
• Lehto O., Virtanen K.I., “Quasiconformal mapping in the plane”, Springer Verlag, Berlin, 1973.
• Mergelyan S. N., “Some questions of constructive functions theory”, Proc. of the Steklov Institute of Mathematics, Vol. XXXVII, 1-92, 1951. [in Russian]
• Özkartepe P., “Pointwise Bernstein-Walsh-type inequalities in regions with piecewise Dini-smooth boundary”, MJEN, 5(3), 35-47 (2017).
• Rickman S., “Characterisation of quasiconformal arcs”, Ann. Acad. Sci. Fenn., Ser. A, Mathematica., 1966, 395 , 30 p.
• Stylianopoulos N., “Strong asymptotics for Bergman polynomials over domains with corners and applications”, Const. Approx., 38, 59-100, (2013).
• Tunc T., Şimşek D., Oruç E., “Pointwise Bernstein-Walsh-type inequalities in regions with interior zero angles in the Bergman space”, Trans. of NAS of Azerbaijan Ser. of Phys.-Tech. and Math. Sci.ences, vol. XXXVII, No 1, 1-12, 2017.
• Walsh J. L., “Interpolation and approximation by rational functions in the complex domain”, AMS, 1960.
• Warschawski S. E., “Über das Randverhalten der Ableitung der Abbildungsfunktion bei konformer Abbildung”, Math.Z., 35, 1932, pp.321-456.