On the growth of derivatives of algebraic polynomials in a weighted Lebesgue space

  • F. G. Аbdullayev Mersin Univ., Turkey and Kyrgyz-Turkish Manas Univ., Bishkek
  • M. Imashkyzy Kyrgyz-Turkish Manas Univ., Bishkek
Keywords: Algebraic polynomial, Quasiconformal mapping, Quasicircle.

Abstract

UDC 517.5

We study growth rates of derivatives of an arbitrary algebraic polynomial in bounded and inbounded regions of the complex plane in weighted Lebesgue spaces.

References

F. G. Abdullayev, V. V. Andrievskii, On the orthogonal polynomials in the domains with K-quasiconformal boundary, Izv. Akad. Nauk Azerb. SSR, Ser. FTM, 1, 3 – 7 (1983).

F. G. Abdullayev, On the some properties on orthogonal polynomials over the regions of complex plane 1, Ukr. Math. J., 52, № 12, 1807 – 1817 (2000), https://doi.org/10.1023/A:1010491406926 DOI: https://doi.org/10.1023/A:1010491406926

F. G. Abdullayev, On the interference of the weight boundary contour for orthogonal polynomials over the region, J. Comput. Anal. and Appl., 6, № 1, 31 – 42 (2004).

F. G. Abdullayev, P. Özkartepe, An analogue of the Bernstein – Walsh lemma in Jordan regions of the complex plane, J. Inequal. and Appl., 2013, Article 570 (2013), https://doi.org/10.1186/1029-242X-2013-570 DOI: https://doi.org/10.1186/1029-242X-2013-570

F. G. Abdullayev, P. Özkartepe, On the behavior of the algebraic polynomial in unbounded regions with piecewise Dini-smooth boundary, Ukr. Math. J., 66, № 5, 645 – 665 (2014), https://doi.org/10.1007/s11253-014-0962-3 DOI: https://doi.org/10.1007/s11253-014-0962-3

F. G. Abdullayev, P. Özkartepe, C. D. Gün, Uniform and pointwise polynomial inequalities in regions without cusps in the weighted Lebesgue space, Bull. TICMI, 18, № 1, 146 – 167 (2014).

F. G. Abdullayev, C. D. Gün, N. P. Özkartepe, Inequalities for algebraic polynomials in regions with exterior cusps, J. Nonlinear Funct. Anal., № 3, 1 – 32 (2015), https://doi.org/10.2298/PIM1614209A

F. G. Abdullayev, P. Özkartepe, Uniform and pointwise polynomial inequalities in regions with cusps in the weighted Lebesgue space, Jaen J. Approx., 7, № 2, 231 – 261 (2015).

F. G. Abdullayev, P. Özkartepe, On the growth of algebraic polynomials in the whole complex plane, J. Korean Math. Soc., 52, № 4, 699 – 725 (2015), https://doi.org/10.4134/JKMS.2015.52.4.699 DOI: https://doi.org/10.4134/JKMS.2015.52.4.699

F. G. Abdullayev, N. P. Özkartepe, Polynomial inequalities in Lavrentiev regions with interior and exterior zero angles in the weighted Lebesgue space, Publ. Inst. Math. (Beograd), 100 (114), № 2, 209 – 227 (2016), https://doi.org/10.2298/PIM1614209A DOI: https://doi.org/10.2298/PIM1614209A

F. G. Abdullayev, N. P. Özkartepe, Interference of the weight and boundary contour for algebraic polynomials in the weighted Lebesgue spaces. I, Ukr. Math. J., 68, № 10, 1574 – 1590 (2017); https://doi.org/10.1007/s11253-017-1313-y DOI: https://doi.org/10.1007/s11253-017-1313-y

F. G. Abdullayev, Polynomial inequalities in regions with corners in the weighted Lebesgue spaces, Filomat, 31, № 18, 5647 – 5670 (2017), https://doi.org/10.2298/fil1718647a DOI: https://doi.org/10.2298/FIL1718647A

F. G. Abdullayev, M. Imashkyzy, G. Abdullayeva, Bernstein – Walsh type inequalities in unbounded regions with piecewise asymptotically conformal curve in the weighted Lebesgue space, J. Math. Sci., 234, № 1, 35 – 48 (2018); https://doi.org/10.1007/s10958-018-3979-6. DOI: https://doi.org/10.1007/s10958-018-3979-6

F. G. Abdullayev, T. Tunç, G. A. Abdullayev, Polynomial inequalities in quasidisks on weighted Bergman space, Ukr. Math. J., 69, № 5, 675 – 695 (2017), https://doi.org/10.1007/s11253-017-1388-5 DOI: https://doi.org/10.1007/s11253-017-1388-5

F. G. Abdullayev, D. ¸Sim¸sek, N. Saypıdınova, Z. Tashpaeva, Polynomial inequalities in regions with piecewise asymptotically conformal curve in the weighted Lebesgue space, Adv. Anal., 3, № 2, 100 – 112 (2018), https://dx.doi.org/10.22606/aan.2018.32004 DOI: https://doi.org/10.22606/aan.2018.32004

F. G. Abdullayev, N. P. Özkartepe, The uniform and pointwise estimates for polynomials on the weighted Lebesgue spaces in the general regions of complex plane, Hacettepe J. Math. and Statist., 48, № 1, 87 – 101 (2019), https://doi.org/10.15672/hjms.2018.639 DOI: https://doi.org/10.15672/HJMS.2018.639

F. G. Abdullayev, P. Özkartepe, T. Tunç, Uniform and pointwise estimates for algebraic polynomials in regions with interior and exterior zero angles, Filomat, 33, № 2, 403 – 413 (2019), https://doi.org/10.2298/fil1902403o DOI: https://doi.org/10.2298/FIL1902403O

F. G. Abdullayev, C. D. Gün, Bernstein – Nikolskii-type inequalities for algebraic polynomials in Bergman space in regions of complex plane, Ukr. Math. J., 73, № 4, 513 – 531 (2021). DOI: https://doi.org/10.1007/s11253-021-01940-z

F. G. Abdullayev, C. D. Gün, Bernstein – Walsh-type inequalities for derivatives of algebraic polynomials, Bull. Korean Math. Soc., 59, № 1, 45 – 72 (2022), https://doi.org/10.4134/BKMS.b210023

F. G. Abdullayev, Bernstein – Walsh-type inequalities for derivatives of algebraic polynomials in quasidiscs, Open Math., 19, № 1, 1847 – 1876 (2021), https://doi.org/10.1515/math-2021-0138 DOI: https://doi.org/10.1515/math-2021-0138

V. V. Andrievskii, V. I. Belyi, V. K. Dzyadyk, Conformal invariants in constructive theory of functions of complex plane, World Federation Publ. Co., Atlanta (1995).

V. V. Andrievskii, Weighted polynomial inequalities in the complex plane, J. Approx. Theory, 164, № 9, 1165 – 1183 (2012), https://doi.org/10.1016/j.jat.2012.05.012 DOI: https://doi.org/10.1016/j.jat.2012.05.012

S. Balcı, M. Imashkyzy, F. G. Abdullayev, Polynomial inequalities in regions with interior zero angles in the Bergman space, Ukr. Math. J., 70, № 3, 362 – 384 (2018), https://doi.org/10.1007/s11253-018-1505-0 DOI: https://doi.org/10.1007/s11253-018-1505-0

D. Benko, P. Dragnev, V. Totik, Convexity of harmonic densities, Rev. Mat. Iberoam, 28, № 4, 1 – 14 (2012), https://doi.org/10.4171/RMI/698 DOI: https://doi.org/10.4171/RMI/698

S. N. Bernstein, Sur la limitation des derivees des polynomes, C. R. Acad. Sci. Paris, 190, 338 – 341 (1930).

P. P. Belinskii, General properties of quasiconformal mappings, Nauka, Sib. otd., Novosibirsk (1974) (in Russian).

V. K. Dzjadyk, Introduction to the theory of uniform approximation of functiıon by polynomials, Nauka, Moskow (1977).

E. Hille, G. Szegö, J. D. Tamarkin, On some generalization of a theorem of A. Markoff, Duke Math. J., 3, 729 – 739 (1937), https://doi.org/10.1215/S0012-7094-37-00361-2 DOI: https://doi.org/10.1215/S0012-7094-37-00361-2

O. Lehto, K. I. Virtanen, Quasiconformal mapping in the plane, Springer-Verlag, Berlin (1973). DOI: https://doi.org/10.1007/978-3-642-65513-5

F. D. Lesley, Holder continuity of conformal mappings at the boundary via the strip method, Indiana Univ. Math. J., 31, 341 – 354 (1982), https://doi.org/10.1512/iumj.1982.31.31030 DOI: https://doi.org/10.1512/iumj.1982.31.31030

D. I. Mamedhanov, Inequalities of S. M. Nikol’skii type for polynomials in the complex variable on curves, Soviet Math. Dokl., 15, 34 – 37 (1974).

G. V. Milovanovic, D. S. Mitrinovic, Th. M. Rassias, Topics in polynomials: extremal problems, inequalities, zeros, World Sci., Singapore (1994), https://doi.org/10.1142/1284 DOI: https://doi.org/10.1142/1284

P. Nevai, V. Totik, Sharp Nikolskii inequalities with exponential weights, Anal. Math., 13, № 4, 261 – 267 (1987), https://doi.org/10.1007/BF01909432 DOI: https://doi.org/10.1007/BF01909432

S. M. Nikol’skii, Approximation of function of several variable and imbeding theorems, Springer-Verlag, New York (1975).

N. P. Özkartepe, C. D. Gün, F. G. Abdullayev, Bernstein – Walsh-type inequalities for derivatives of algebraic polynomials on the regions of complex plane (to appear).

Ch. Pommerenke, Univalent functions, Vandenhoeck & Ruprecht, Gottingen (1975).

Ch. Pommerenke, Boundary behaviour of conformal maps, Springer-Verlag, Berlin (1992). DOI: https://doi.org/10.1007/978-3-662-02770-7

I. Pritsker, Comparing norms of polynomials in one and several variables, J. Math. Anal. and Appl., 216, 685 – 695 (1997); https://doi.org/10.1006/jmaa.1997.5699. DOI: https://doi.org/10.1006/jmaa.1997.5699

N. Stylianopoulos, Strong asymptotics for Bergman polynomials over domains with corners and applications, Const. Approx., 38, № 1, 59 – 100 (2012), https://doi.org/10.1007/s00365-012-9174-y DOI: https://doi.org/10.1007/s00365-012-9174-y

G. Szego, A. Zygmund, On certain mean values of polynomials, J. Anal. Math., 3, № 1, 225 – 244 (1953), https://doi.org/10.1007/BF02803592 DOI: https://doi.org/10.1007/BF02803592

S. E. Warschawski, On Holder continuity at the boundary in conformal maps, J. Math. and Mech., 18, 423 – 427 (1968), https://doi.org/10.1512/iumj.1969.18.18032 DOI: https://doi.org/10.1512/iumj.1969.18.18032

J. L. Walsh, Interpolation and approximation by rational functions in the complex domain, Amer. Math. Soc. (1960).

Published
17.06.2022
How to Cite
АbdullayevF. G., and M. Imashkyzy. “On the Growth of Derivatives of Algebraic Polynomials in a Weighted Lebesgue Space”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 5, June 2022, pp. 582 -98, doi:10.37863/umzh.v74i5.7052.
Section
Research articles