Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On inequalities for differential operators

Author: Richard Bellman
Journal: Proc. Amer. Math. Soc. 9 (1958), 589-597
MSC: Primary 26.00
MathSciNet review: 0110771
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we study the following problem: Given that certain functionals of $ u$ and its derivatives belong to given $ {\text{L}}$classes over the infinite interval, what can be said about the $ {\text{L}}$-classes of other functionals? Utilizing a simple device from the theory of linear differential equations, we obtain a number of results due to Landau, Kolmogoroff, Halperin-von Neumann, and Nagy, together with some extensions.

References [Enhancements On Off] (What's this?)

  • [1] R. Bellman, An integral inequality, Duke Math. J. vol. 10 (1943) pp. 547-550. MR 0008416 (5:1b)
  • [2] -, Stability theory of differential equations, New York, McGraw-Hill, 1954.
  • [3] R. P. Boas, Entire functions, New York, Academic Press, 1954. MR 0068627 (16:914f)
  • [4] G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Cambridge University Press, 1934.
  • [5] I. Halperin, Closures and adjoints of linear differential operators, Ann. of Math. vol. 38 (1937) pp. 880-919. MR 1503380
  • [6] A. Kolmogoroff, Une generalisation de l'inegalité de M. J. Hadamard entre les bornes supérieures des dérivées successive d'une fonction, C. R. Acad. Sci. Paris, vol. 207 (1938) pp. 764-765.
  • [7] E. Landau, Math. Ann. vol. 102 (1929) pp. 177-178.
  • [8] V. Levin, Appendix of Russian edition of the book by Hardy, Littlewood and Polya referred to above.
  • [9] B. Nagy, Uber Integralungleichungen zwischen einer Funktion und ihrer Ableitungen, Acta Sci. Math. Szeged. vol. 10 (1941) pp. 64-74. MR 0004277 (2:351b)
  • [10] E. M. Stein, Functions of exponential type, Ann. of Math. vol. 65 (1957) pp. 582-592. MR 0085342 (19:23i)
  • [11] I. Halperin and H. R. Pitt, Integral inequalities connected with differential operators, Duke Math. J. vol. 4 (1938) pp. 613-625. MR 1546080
  • [A] C. Schaeffer, Inequalities of A. Markoff and S. Bernstein for polynomials and related functions, Bull. Amer. Math. Soc. vol. 47 (1941) pp. 565-579. MR 0005163 (3:111a)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 26.00

Retrieve articles in all journals with MSC: 26.00

Additional Information

Article copyright: © Copyright 1958 American Mathematical Society

American Mathematical Society