We establish Bernstein-type inequalities for the fractional integroderivatives of arbitrary algebraic polynomials in the space L 0 .
References
K. Mahler, “On the zeros of the derivative of a polynomial,” Proc. Roy. Soc. London A, 264, 145–154 (1961).
T. M. Flett, “Mean values of power series,” Pacif. J. Math., 25, 163–194 (1968).
T. M. Flett, “Temperatures Bessel potentials and Lipschitz spaces,” Proc. London Math. Soc., 22, No. 3, 385–481 (1971).
V. V. Arestov, “Integral inequalities for algebraic polynomials on the unit circle,” Mat. Zametki, 48, No. 4, 7–18 (1990).
É. A. Storozhenko, “On one Mahler problem of zeros of a polynomial and its derivative,” Mat. Sb., 187, No. 5, 111–120 (1996).
G. Pólya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis. Ester Band. Reihen. Integralrechnung. Funktionentheorie [Russian translation], Nauka, Moscow (1978).
N. J. de Bruijn and T. A. Springer, “On the zeros of composition polynomials,” Jndag. Math., 9, 406–414 (1947).
V. V. Arestov, “On the integral inequalities for trigonometric polynomials and their derivatives,” Izv. Akad. Nauk SSSR, Ser. Mat., 45, 3–22 (1981).
I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products [in Russian], Gostekhteorizdat, Moscow (1963).
É. A. Storozhenko and L. G. Kovalenko, “Inequalities for fractional integrals of complex polynomials in L 0 ,” Mat. Zametki, 96, No. 4, 633–636 (2014).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 5, pp. 705–710, May, 2017.
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Kovalenko, L.G., Storozhenko, É.A. On the Fractional Integrodifferentiation of Complex Polynomials in L 0 . Ukr Math J 69, 823–830 (2017). https://doi.org/10.1007/s11253-017-1398-3
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DOI: https://doi.org/10.1007/s11253-017-1398-3