The well-known Taikov’s refined versions of the Hardy – Littlewood – Pólya inequality for the L 2 -norms of intermediate derivatives of a function defined on the real axis are generalized to the case of powers of self-adjoint operators in a Hilbert space.
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G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge University, Cambridge (1934).
V. F. Babenko, N. P. Korneichuk, V. A. Kofanov, and S. A. Pichugov, Inequalities for Derivatives and Their Applications [in Russian], Naukova Dumka, Kiev (2003).
L. V. Taikov, “Refinement of a Hardy inequality that contains an estimate for the value of the intermediate derivative of a function,” Mat. Zametki, 50, No. 4, 114–122 (1991).
L. V. Taikov, “On Hardy inequalities,” Mat. Zametki, 52, No. 4, 106–111 (1992).
N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in a Hilbert Space [in Russian], Nauka, Moscow (1966).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 10, pp. 1299 – 1305, October, 2009.
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Babenko, V.F., Bilichenko, R.O. Refinement of a Hardy–Littlewood–Pólya-type inequality for powers of self-adjoint operators in a Hilbert space. Ukr Math J 61, 1533–1540 (2009). https://doi.org/10.1007/s11253-010-0295-9
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DOI: https://doi.org/10.1007/s11253-010-0295-9