Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-12-01T01:39:03.590Z Has data issue: false hasContentIssue false
  • English
  • Français

Ramifications of Landau's inequality

Published online by Cambridge University Press:  14 November 2011

Man Kam Kwong  and
A. Zettl
Man Kam Kwong
Affiliation:
Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115, U.S.A.
A. Zettl
Affiliation:
Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Synopsis

The problem of determining the best constant κ in the inequality ‖y′‖≦K ‖y‖ ‖y″‖ is discussed in the context of the classical Lp spaces, 1 ≦ p ≦ ∞.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 86 , Issue 3-4 , 1980 , pp. 175 - 212
Copyright
Copyright © Royal Society of Edinburgh 1980

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

References and Bibliography

1Adams, R. A.. Sobolev Spaces (New York: Academic Press, 1975).Google Scholar
2Arestov, V. V.. Exact inequalities between norms of functions and their derivatives. Acta Sci. Math. 33 (1972), 243–267.Google Scholar
3Arestov, V. V.. The best approximation to differentiation operators. Mat. Zametki 1 (1967), pt 2; 149–154.Google Scholar
4Arestov, V. V.. On the best approximation of the operators of differentiation and related questions. Approximation Theory, Proceeding of Conference in Poznan, Poland 1972 (Boston: Reidel, 1975).Google Scholar
5Berdyshev, V. I.. The best approximation in L(0, ∞) to the differentiation operator. Mat. Zametki 5 (1971), 477–481.Google Scholar
6Bosse, Yu. G. (Shilov, G. E.). O neravenstvakh mezhdu proizvodnymi. Mosk. Univ. Sbornik raport nauchnykh studencheskikh kruzhkov, 1937, 17–27.Google Scholar
7Bradley, J. and Everitt, W. N.. On the inequality ∥f“∥2 ≦ K ∥f∥ ∥f (4)∥ Quart. J. Math. 25 (1974), 241–252.CrossRefGoogle Scholar
8Bradley, J. and Everitt, W. N.. Inequalities associated with regular and singular problems in the calculus of variations. Trans. Amer. Math. Soc. 182 (1973), 303–321.CrossRefGoogle Scholar
9Brodlie, K. W. and Everitt, W. N.. On an inequality of Hardy and Littlewood. Proc. Roy. Soc. Edinburgh Sect. A 72 (1975), 179–186.Google Scholar
10Cavaretta, A. S.. An elementary proof of Kolmogorov's theorem. Amer Math. Monthly 81 (1974), 480–486.CrossRefGoogle Scholar
11Cavaretta, A. S.. One-sided inequalities for the successive derivatives of a function. Bull Amer. Math. Soc. 82 (1976), 303–305.CrossRefGoogle Scholar
12Cavaretta, A. S.. A refinement of Kolmogorov's inequality. MRC Technical Summary Report 1788 (1977).Google Scholar
13Cartan, H.. On inequalities between the maxima of the successive derivatives of a function. C. R. Acad. Sci. Paris 208 (1939), 414–426.Google Scholar
14Certain, M. W. and Kurtz, T. G.. Landau-Kolmogorov inequalities for semigroups and groups. Proc. Amer. Math. Soc. 63 (1977), 226–230.CrossRefGoogle Scholar
15Ciesielski, Z. and Musielak, J.. Approximation Theory. Proceedings of the Conference jointly organized by the Mathematical Institute of the Polish Academy of Sciences and the Institute of Mathematics of the Adam Mickiewicz University held in Proznan 22–26 August, 1972. (Boston: Reidel, 1975).Google Scholar
16Copson, E. T.. On two integral inequalities. Proc. Roy. Soc. Edinburgh. Sect. A 77 (1977), 325–328.Google Scholar
17Copson, E. T.. On two inequalities of Brodlie and Everitt. Proc. Roy. Soc. Edinburgh Sect. A 77 (1977), 329–333.Google Scholar
18Ditzian, Z.. Some remarks on inequalities of Landau and Kolmogorov Aequationes Math. 12 (1975), 145–151.CrossRefGoogle Scholar
19Ditzian, Z.. Note on Hille's question. Aequationes Math. 15 (1977), 143–144.CrossRefGoogle Scholar
20Evans, W. D. and Zettl, A.. Norm inequalities involving derivatives. Proc. Rov. Soc. Edinburgh Sect. A 82 (1978), 51–70.Google Scholar
21Everitt, W. N.. On an extension to an integro-differential inequality of Hardy, Littlewood and Polya. Proc. Roy. Soc. Edinburgh Sect. A 69 (1972), 295–333.Google Scholar
22Everitt, W. N. and Giertz, M.. On the integro-differential inequality J. Math. Anal. Appl. 45 (1974), 639–653.CrossRefGoogle Scholar
23Everitt, W. N. and Giertz, M.. Some inequalities associated with certain ordinary differential operators. Math. Z. 126 (1972), 308–326.CrossRefGoogle Scholar
24Everitt, W. N. and Giertz, M.. Inequalities and separation for certain ordinary differential operators. Proc. London Math. Soc. 28 (1974), 352–372.CrossRefGoogle Scholar
25Everitt, W. N. and Jones, D. S.. On an integral inequality. Proc. Roy. Soc. London Ser. A 357 (1977), 271–288.Google Scholar
26Everitt, W. N. and Zettl, A.. On a class of integral inequalities. J. London Math. Soc. 17 (1978), 291–303.CrossRefGoogle Scholar
27Friedman, A.. Partial Differential Equations (New York: Holt, Rinehart and Winston, 1969).Google Scholar
28Fink, A. M.. Best possible approximation constants. Trans. Amer. Math. Soc. 226 (1977), 243–255.CrossRefGoogle Scholar
29Gabushin, V. N.. Inequalities for norms of a function and its derivatives in L p metrics. Mat. Zametki 1 (1967), 291–8.Google Scholar
30Gabushin, V. N.. Exact constants in inequalities between norms of derivatives of function. Mat. Zametki 4 (1968), 221–32.Google Scholar
31Gabushin, V. N.. The best approximation for differentiation operators on the half-line. Mat. Zametki 6 (1969), 573–82.Google Scholar
32Gindler, H. A. and Goldstein, J. A.. Dissipative operator versions of some classical inequalities. J. Analyse Math. 28 (1975), 213–238.CrossRefGoogle Scholar
33Goldstein, J.. On improving the constants in the Kolmogorov inequalities, preprint.Google Scholar
34Gorny, A.. Contributions to the study of differentiable functions of a real variable. Acta Math. 71 (1939), 317–358.CrossRefGoogle Scholar
35Hadamard, J.. Sur le module maximum d'une fonction et de ses dérivées. C R. Soc. Math. France (1914), 68–72.Google Scholar
36Hardy, G. H. and Littlewood, J. E.. Some integral inequalities connected with the calculus of variations. Quart. J. Math. Oxford Ser. 3 (1932), 241–252.CrossRefGoogle Scholar
37Hardy, G. H., Littlewood, J. E. and Polya, G.. Inequalities (Cambridge Univ. Press, 1934).Google Scholar
38Hille, E.. Generalizations of Landau's inequality to linear operators. In Linear operators and approximation (Butzer, P. L.et. al.). (Basel and Stuttgart. Birkhäuser Verlag, 1972).Google Scholar
39Hille, E.. Remark on the Landau-Kallman-Rota inequality. Aequationes Math. 4 (1970), 239–240.Google Scholar
40Hille, E.. On the Landau-Kallman-Rota inequality. J. Approximation Theory 6 (1972), 117–122.CrossRefGoogle Scholar
41Hille, E. and Phillips, R. S.. Functional analysis and semigroups. Amer. Math. Soc. Coll. Publ. 31, Rev. ed. (Providence, R.I.: A.M.S., 1957).Google Scholar
42Holbrook, J. A. R.. A Kallman-Rota-Kato inequality for nearly euclidean spaces, preprint.Google Scholar
43Kallman, R. R. and Rota, G. C.. On the inequality ∥f′∥2≦4 ∥f∥ ∥f“∥. In Inequalities II (ed. Shisha, O.) 187–192, (New York Academic Press, 1970).Google Scholar
44Kato, T.. On an inequality of Hardy, Littlewood and Polya Advances in Math. 7 (1971), 217–218.CrossRefGoogle Scholar
45Kolmogorov, A. N.. On inequalities between the upper bounds of the successive derivatives of an arbitrary function on an infinite interval. Amer. Math. Soc. Transi. (1) 2 (1962), 233–243.Google Scholar
46Kurepa, S.. Remarks on the Landau inequality. Aequations Math. 4 (1970), 240–241.Google Scholar
47Kwong, M. K. and Zettl, A.. Remarks on best constants for norm inequalities among powers of an operator. J. Approximation Theory 3 (1979), 249–258.CrossRefGoogle Scholar
48Landau, E.. Einige Ungleichungen für zweimal differenzierbare Funktionen. Proc. London Math. Soc. 13 (1913), 43–49.Google Scholar
49Ljubič, Ju. I.(or Yu Lyubich). On inequalities between the powers of a linear operator. Transi Amer. Math. Soc. (2) 40 (1964), 39–84; translated from Izv. Akad. Nauk. SSSR Ser. Mat. 24 (1960), 825–864.Google Scholar
50Matorin, A. P.. Inequalities between the maximum absolute values of a function and its derivatives on the half-line. Ukrain. Mat. Z. 7 (1955), 262–266.Google Scholar
51Mitrinovic, D. S.. Analytic Inequalities (Berlin: Springer-Verlag, 1970).CrossRefGoogle Scholar
52Nirenberg, L.. Remarks on strongly elliptic partial differential equations. Comm. Pure Appl. Math., 8 (1955), 648–674.CrossRefGoogle Scholar
53Nagy, B. Sz.. Über Integralungleichungen zwischen einer Funktion und ihrer Ableitung. Acta Sci. Math. 10 (1941), 64–74.Google Scholar
54Redhefier, R. M.. Über eine beste Ungleichung zwischen den Normen von ƒ, ƒ′, ƒ”. Math. Z. 80 (1963), 390–397.CrossRefGoogle Scholar
55Schonberg, I. J.. The elementary cases of Landau's problem of inequalities between derivatives, Amer. Math. Monthly 80 (1973), 121–158.CrossRefGoogle Scholar
56Schonberg, I. J., and Cavaretta, A.. Solution of Landau's problem concerning higher derivatives on the half line. MRCT. S. R. 1050, Madison, Wisconsin, 1970.Google Scholar
57Stechkin, S. B.. Inequalities between norms of derivatives of an arbitrary function. Acta. Sci. Math. 26 (1965), 225–230.Google Scholar
58Stechkin, S. B.. The inequalities between upper bounds for the derivatives of an arbitrary function on the half-line. Mat. Zametki 1 (1967), 665–674.Google Scholar
59Stein, E. M.. Functions of exponential type. Ann. of Math. (2) 65 (1957), 582–592.CrossRefGoogle Scholar
60Trebels, W. and Westphal, V. I.. A note on the Landau-Kallman-Rota-Hille inequality. Linear Operators and Approximation (ed. Butzer, P. L.et. al) (Basel: Birkhäuser, 1972).Google Scholar
61Taikov, L. V.. Inequalities of Kolmogorov type and the best formulae for numerical differentiation. Mat. Zametki 4 (1968), 233–238.Google Scholar