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On the Growth of Derivatives of Algebraic Polynomials in a Weighted Lebesgue Space

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Ukrainian Mathematical Journal Aims and scope

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

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References

  1. F. G. Abdullayev and V. V. Andrievskii, “On the orthogonal polynomials in the domains with K–quasiconformal boundary,” Izv. Akad. Nauk Azerbaĭdzhan. SSR Ser. Fiz.-Tekh. Mat. Nauk, 1, 3–7 (1983).

    MathSciNet  Google Scholar 

  2. F. G. Abdullayev, “On some properties on orthogonal polynomials over the regions of complex plane 1,” Ukr. Math. Zh., 52, No. 12, 1587–1595 (2000); English translation: Ukr. Math. J., 52, No. 12, 1807–1817 (2000).

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

    MathSciNet  MATH  Google Scholar 

  4. F. G. Abdullayev and P. Özkartepe, “An analogue of the Bernstein–Walsh lemma in Jordan regions of the complex plane,” J. Inequal. Appl., 2013, Article 570 (2013).

  5. F. G. Abdullayev and P. Özkartepe, “On the behavior of the algebraic polynomial in unbounded regions with piecewise Dini-smooth boundary,” Ukr. Math. Zh., 66, No. 5, 579–597 (2014); English translation: Ukr. Math. J., 66, No. 5, 645–665 (2014).

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

    MathSciNet  MATH  Google Scholar 

  7. F. G. Abdullayev, C. D. Gün, and N. P. Ozkartepe, “Inequalities for algebraic polynomials in regions with exterior cusps,” J. Nonlin. Funct. Anal., No. 3, 1–32 (2015).

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

    MathSciNet  MATH  Google Scholar 

  9. F. G. Abdullayev and P. Özkartepe, “On the growth of algebraic polynomials in the whole complex plane,” J. Korean Math. Soc., 52, No. 4, 699–725 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  10. F. G. Abdullayev and 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), No. 2, 209–227 (2016).

  11. F. G. Abdullayev and N. P. Özkartepe, “Interference of the weight and boundary contour for algebraic polynomials in the weighted Lebesgue spaces. I,” Ukr. Math. Zh., 68, No. 10, 1365–1379 (2017); English translation: Ukr. Math. J., 68, No. 10, 1574–1590 (2017); DOI:https://doi.org/10.1007/s11253-017-1313-y.

  12. F. G. Abdullayev, “Polynomial inequalities in regions with corners in the weighted Lebesgue spaces,” Filomat, 31, No. 18, 5647–5670 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  13. F. G. Abdullayev, M. Imashkyzy, and G. Abdullayeva, “Bernstein–Walsh type inequalities in unbounded regions with piecewise asymptotically conformal curve in the weighted Lebesgue space,” Ukr. Mat. Visn., 14, No. 4, 515–531 (2017); English translation: J. Math. Sci. (N.Y.), 234, No. 1, 35–48 (2018); DOI: 10.1007/s10958-018-3979-6.

  14. F. G. Abdullayev, T. Tunc, and G. A. Abdullayev, “Polynomial inequalities in quasidisks on weighted Bergman space,” Ukr. Math. Zh., 69, No. 5, 582–598 (2017); English translation: Ukr. Math. J., 69, No. 5, 675–695 (2017).

  15. F. G. Abdullayev, D. Şimşek, N. Saypıdınova, and Z. Tashpaeva, “Polynomial inequalities in regions with piecewise asymptotically conformal curve in the weighted Lebesgue space,” Adv. Anal., 3, No. 2, 100–112 (2018).

    Article  Google Scholar 

  16. F. G. Abdullayev and N. P. Özkartepe, “The uniform and pointwise estimates for polynomials on the weighted Lebesgue spaces in the general regions of complex plane,” Hacet. J. Math. Stat., 48, No. 1, 87–101 (2019).

    MathSciNet  MATH  Google Scholar 

  17. F. G. Abdullayev, P. Özkartepe, and T. Tunç, “Uniform and pointwise estimates for algebraic polynomials in regions with interior and exterior zero angles,” Filomat, 33, No. 2, 403–413 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  18. F. G. Abdullayev and C. D. Gün, “Bernstein–Nikolskii-type inequalities for algebraic polynomials in Bergman space in regions of complex plane,” Ukr. Math. Zh., 73, No. 4, 439–454 (2021); English translation: Ukr. Math. J., 73, No. 4, 513–531 (2021).

  19. F. G. Abdullayev and C. D. Gün, “Bernstein–Walsh-type inequalities for derivatives of algebraic polynomials,” Bull. Korean Math. Soc., 59, No. 1, 45–72 (2022).

    MathSciNet  MATH  Google Scholar 

  20. F. G. Abdullayev, “Bernstein–Walsh-type inequalities for derivatives of algebraic polynomials in quasidiscs,” Open Math., 19, No. 1, 1847–1876 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  21. V. V. Andrievskii, V. I. Belyi, and V. K. Dzyadyk, Conformal Invariants in Constructive Theory of Functions of Complex Plane, World Federation Publishers Company, Atlanta, GA (1995).

  22. V. V. Andrievskii, “Weighted polynomial inequalities in the complex plane,” J. Approx. Theory, 164, No. 9, 1165–1183 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  23. S. Balcı, M. Imashkyzy, and F. G. Abdullayev, “Polynomial inequalities in regions with interior zero angles in the Bergman space,” Ukr. Math. Zh., 70, No. 3, 318–336 (2018); English translation: Ukr. Math. J., 70, No. 3, 362–384 (2018).

  24. D. Benko, P. Dragnev, and V. Totik, “Convexity of harmonic densities,” Rev. Mat. Iberoam, 28, No. 4, 1–14 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  25. S. N. Bernstein, “Sur la limitation des dérivées des polynômes,” C. R. Acad. Sci. Paris, 190, 338–341 (1930).

    MATH  Google Scholar 

  26. P. P. Belinskii, General Properties of Quasiconformal Mappings [in Russian], Nauka, Sib. otd., Novosibirsk (1974).

  27. V. K. Dzyadyk, Introduction to the Theory of Uniform Approximation of Functions by Polynomials, Nauka, Moscow (1977).

    MATH  Google Scholar 

  28. E. Hille, G. Szegö, and J. D. Tamarkin, “On some generalization of a theorem of A. Markoff,” Duke Math. J., 3, 729–739 (1937).

    Article  MathSciNet  MATH  Google Scholar 

  29. O. Lehto and K. I. Virtanen, Quasiconformal Mapping in the Plane, Springer, Berlin (1973).

    Book  MATH  Google Scholar 

  30. F. D. Lesley, “Hölder continuity of conformal mappings at the boundary via the strip method,” Indiana Univ. Math. J., 31, 341–354 (1982).

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  32. G. V. Milovanovic, D. S. Mitrinovic, and Th. M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific, Singapore (1994).

  33. P. Nevai and V. Totik, “Sharp Nikolskii inequalities with exponential weights,” Anal. Math., 13, No. 4, 261–267 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  34. S. M. Nikol’skii, Approximation of Function of Several Variable and Imbedding Theorems, Springer-Verlag, New York (1975).

    Book  Google Scholar 

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

  36. Ch. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen (1975).

    MATH  Google Scholar 

  37. Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin (1992).

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  39. N. Stylianopoulos, “Strong asymptotics for Bergman polynomials over domains with corners and applications,” Constr. Approx., 38, No. 1, 59–100 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  40. G. Szegö and A. Zygmund, “On certain mean values of polynomials,” J. Analyse Math., 3, No. 1, 225–244 (1953).

    Article  MathSciNet  MATH  Google Scholar 

  41. S. E. Warschawski, “On Hölder continuity at the boundary in conformal maps,” J. Math. Mech., 18, 423–427 (1968).

    MathSciNet  MATH  Google Scholar 

  42. J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, American Mathematical Society, Providence, R.I. (1960).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Mersin University, Turkey and Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic

    F. G. Abdullayev

  2. Kyrgyz-Turkish Manas University, Bishkek, Kyrgyz Republic

    M. Imashkyzy

Authors
  1. F. G. Abdullayev
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  2. M. Imashkyzy
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Correspondence to F. G. Abdullayev.

Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 5, pp. 582–598, May, 2022. Ukrainian DOI: https://doi.org/10.37863/umzh.v74i5.7052.

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Abdullayev, F.G., Imashkyzy, M. On the Growth of Derivatives of Algebraic Polynomials in a Weighted Lebesgue Space. Ukr Math J 74, 664–684 (2022). https://doi.org/10.1007/s11253-022-02093-3

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  • DOI: https://doi.org/10.1007/s11253-022-02093-3

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