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Hadamard–Landau inequalities in uniformly convex spaces

Published online by Cambridge University Press:  24 October 2008

J. R. Partington
J. R. Partington
Affiliation:
Pembroke College, Cambridge
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Extract

The inequality

for fεLp(− ∞, ∞)or Lp(0, ∞) (1≤p ≤ ∞), and its extension

for T an Hermitian or dissipative linear operator, in general unbounded, on a Banach space X, for xεX, have been considered by many authors. In particular, forms of inequality (1) have been given by Hadamard(7), Landau(15), and Hardy and Little-wood(8),(9). The second inequality has been discussed by Kallman and Rota(11), Bollobás (2) and Kato (12), and numerous further references may be found in the recent papers of Kwong and Zettl(i4) and Bollobás and Partington(3).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 90 , Issue 2 , September 1981 , pp. 259 - 264
Copyright
Copyright © Cambridge Philosophical Society 1981

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References

REFERENCES

(1)Berdyshev, V. I.The best approximation in L(0, ∞) to the differentiation operator. Mat. Zametki 5 (1971), 477–481.Google Scholar
(2)Bollosás, B.The spatial numerical range and powers of an operator. J. London Math. Soc. 7 (1973), 435–440.Google Scholar
(3)Bollobás, B. and Partington, J. R.Inequalities for quadratic polynomials in Hermitian and dissipative operators. Advances in Math, (to appear).Google Scholar
(4)Bonsall, F. F. and Duncan, J. Numerical ranges of operators on normed spaces and elements of normed algebras. London Math. Soc. Lecture Notes 2 (Cambridge University Press, 1971).Google Scholar
(5)Bonsall, F. F. and Duncan, J.Numerical ranges II. London Math. Soc. Lecture Notes 10 (Cambridge University Press, 1973).Google Scholar
(6)Day, M. M.Normed Linear Spaces (Springer, 3rd edition, 1972).Google Scholar
(7)Hadamard, J.Sur le module maximum d'une fonction et de ses dérivées. Bull. Soc. Math. France 42, C.R. Séances (1914), 68–72.Google Scholar
(8)Hardy, G. H. and Littlewood, J. E.Some integral inequalities connected with the calculus of variations. Quart. J. Math. 3 (1932), 241–252.Google Scholar
(9)Hardy, G. H., Littlewood, J. E. and Polya, G.Inequalities (Cambridge University Press, 1952).Google Scholar
(10)Holbrooke, J. A. R.A Kallman-Rota inequality for nearly Euclidean spaces. Advances in Math. 14 (1974), 335–345.Google Scholar
(11)Kallman, R. R. and Rota, G.-C. On the inequality ∣f'∣2.≤ 4 ∣f∣ ∣f"∣ Inequalities, vol. II, ed. Shisha, O. (Academic Press, 1970), pp. 187–192.Google Scholar
(12)Kato, T.On an inequality of Hardy, Littlewood and Polya. Advances in Math. 7 (1971), 217–218.Google Scholar
(13)Kwong, M. K. and Zettl, A.Remarks on best constants for norm inequalities among powers of an operator. J. Approx. Theory 26 (1979), 249–258.Google Scholar
(14)Kwong, M. K. and Zettl, A.Ramifications of Landau's inequality. Proc. Roy. Soc. Edinburgh 86 A (1980), 175–212.CrossRefGoogle Scholar
(15)Landau, E.Einige Ungleichungen für zweimal differentierbare Funktionen. Proc. London Math. Soc. 13 (1914), 43–49.CrossRefGoogle Scholar
(16)Lewis, D. R.Finite dimensional subspaces of Lp. Studia Math. 63 (1978), 207–212.Google Scholar
(17)Partington, J. R.Constants relating a Hermitian operator and its square. Math. Proc. Cambridge Philos. Soc. 85 (1979), 325–333.CrossRefGoogle Scholar