Нерiвностi типу Маркова–Нiкольського в областях iз внутрiшнiми нульовими кутами в просторi Бергмана

P.  Özkartepe

doi:10.37863/umzh.v75i3.7322

P. Özkartepe Gaziantep University

DOI: https://doi.org/10.37863/umzh.v75i3.7322

Abstract

UDC 517.5

The order of growth of the module of an arbitrary algebraic polynomial in a weighted Bergman space $A_{p}(G,h),$ $p>0,$ is investigated in the regions with exterior nonzero and interior zero angles at finitely many points of the boundary. We establish estimates of the Markov–,Nikolskii type for algebraic polynomials and clarify the behavior of derived polynomials at the points of zeros and poles of the weight function in bounded regions with piecewise-smooth boundary.

References

F. G. Abdullayev, V. V. Andrievskii, On the orthogonal polynomials in the domains with $K$-quasiconformal boundary, (in Russian), Izv. Akad. Nauk Azerb. SSR, Ser. Fiz., Tech., Mat., 4, № 1, 7–11 (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).

F. G. Abdullayev, On the some properties of the orthogonal polynomials over the region of the complex plane (Part III), Ukr. Math. J., 53, № 12, 1934–1948 (2001).

F. G. Abdullayev, U. Deger, On the orthogonal polynomials with weight having singularities on the boundary of regions in the complex plane, Bull. Belg. Math. Soc., 16, № 2, 235–250 (2009). DOI: https://doi.org/10.36045/bbms/1244038136

F. G. Abdullayev, N. D. Aral, On Bernstein–Walsh-type lemmas in regions of the complex plane, Ukr. Math. J., 63, № 3, 337–350 (2011). DOI: https://doi.org/10.1007/s11253-011-0507-y

F. G. Abdullayev, C. D. Gün, On the behavior of the algebraic polynomials in regions with pie-cewise smooth boundary without cusps, Ann. Polon. Math., 111, 39–58 (2014). DOI: https://doi.org/10.4064/ap111-1-4

F. G. Abdullayev, N. P. Özkartepe, On the behavi or of the algebrai c polynomi al in unbounded regi ons wi th pi ecewi se Di ni–Smooth boundary, Ukr. Math. J., 66, № 5, 579–597 (2014). DOI: https://doi.org/10.1007/s11253-014-0962-3

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). DOI: https://doi.org/10.4134/JKMS.2015.52.4.699

F. G. Abdullayev, N. P. Özkartepe, Uniform and pointwise Bernstein–Walsh-type inequalities on a quasidisk in the complex plane, Bull. Belg. Math. Soc., 23, № 2, 285–310 (2016). DOI: https://doi.org/10.36045/bbms/1464710119

F. G. Abdullayev, T. Tunc{c, Uniform and pointwise polynomial inequalities in regions with asymptotically conformal curve on weighted Bergman space, Lobachevskii J. Math., 38, № 2, 193–205 (2017). DOI: https://doi.org/10.1134/S1995080217020020

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

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); DOI.org/10.4134/BKMS.b210023.

F. G. Abdullayev, Bernstein–Walsh-type inequalities for derivatives of algebraic polynomials in quasidiscs, Open Math., 19, 1847–1876 (2021). DOI: https://doi.org/10.1515/math-2021-0138

L. Ahlfors, Lectures on quasiconformal mappings, Van Nostrand, Princeton, NJ (1966).

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, H. P. Blatt, Discrepancy of signed measures and polynomial approximation, Springer-Verlag, New York (2010).

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

S. Balci, 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). 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). DOI: https://doi.org/10.4171/RMI/698

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

S. N. Bernstein, On the best approximation of continuos functions by polynomials of given degree, Izd. Akad. Nauk SSSR, I (1952); II (1954) (O nailuchshem priblizhenii nepreryvnykh funktsii posredstrvom mnogochlenov dannoi stepeni), Sobraniye sochinenii, I (4), 11–10 (1912).

V. K. Dzyadyk, I. A. Shevchuk, Theory of uniform approximation of functions by polynomials, Walter de Gruyter, Berlin, New York (2008). DOI: https://doi.org/10.1515/9783110208245

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

D. Jackson, Certain problems on closest approximations, Bull. Amer. Math. Soc., 39, 889–906 (1933). DOI: https://doi.org/10.1090/S0002-9904-1933-05759-2

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

D. I. Mamedhanov, Inequalities of S. M. Nikol'skii type for polynomials in the complex variable on curves, Soviet Mat. 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). DOI: https://doi.org/10.1142/1284

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

P. Özkartepe, Uniform and pointwise polynomial estimates in regions with interior and exterior cusps, Cumhuriyet Sci. J., 39, № 1, 47–65 (2018). DOI: https://doi.org/10.17776/csj.405512

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

Ch. Pommerenke, Univalent functions, Vandenhoeck and Ruprecht, Göttingen (1975).

S. Rickman, Characterization of quasiconformal arcs, Ann. Acad. Sci. Fenn., Math., 395, 7–30 (1966). DOI: https://doi.org/10.5186/aasfm.1967.395

P. M. Tamrazov, Smoothness and polynomial approximations} (in Russian), Naukova Dumka, Kiev (1975).

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

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

Inequalities of the Markov–Nikolskii type in regions with zero interior angles in Bergman space

Abstract

References