Some quadratic integral inequalities of Opial type

Małgorzata Kuchta

Annales Polonici Mathematici (1996)

  • Volume: 63, Issue: 2, page 103-113
  • ISSN: 0066-2216

Abstract

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We derive and investigate integral inequalities of Opial type: ∫ I s | h h ̇ | d t ≤ ∫ I r h ̇ ² d t , where h ∈ H, I = (α,β) is any interval on the real line, H is a class of absolutely continuous functions h satisfying h(α) = 0 or h(β) = 0. Our method is a generalization of the method of [3]-[5]. Given the function r we determine the class of functions s for which quadratic integral inequalities of Opial type hold. Such classes have hitherto been described as the classes of solutions of a certain differential equation. In this paper a wider class of functions s is given which is the set of solutions of a certain differential inequality. This class is determined directly and some new inequalities are found.

How to cite

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Małgorzata Kuchta. "Some quadratic integral inequalities of Opial type." Annales Polonici Mathematici 63.2 (1996): 103-113. <http://eudml.org/doc/262697>.

@article{MałgorzataKuchta1996,
abstract = {We derive and investigate integral inequalities of Opial type: $∫_I s|hḣ|dt ≤ ∫_I rḣ² dt$, where h ∈ H, I = (α,β) is any interval on the real line, H is a class of absolutely continuous functions h satisfying h(α) = 0 or h(β) = 0. Our method is a generalization of the method of [3]-[5]. Given the function r we determine the class of functions s for which quadratic integral inequalities of Opial type hold. Such classes have hitherto been described as the classes of solutions of a certain differential equation. In this paper a wider class of functions s is given which is the set of solutions of a certain differential inequality. This class is determined directly and some new inequalities are found.},
author = {Małgorzata Kuchta},
journal = {Annales Polonici Mathematici},
keywords = {absolutely continuous function; integral inequality; absolutely continuous functions; quadratic integral inequalities of Opial type; differential inequality},
language = {eng},
number = {2},
pages = {103-113},
title = {Some quadratic integral inequalities of Opial type},
url = {http://eudml.org/doc/262697},
volume = {63},
year = {1996},
}

TY - JOUR
AU - Małgorzata Kuchta
TI - Some quadratic integral inequalities of Opial type
JO - Annales Polonici Mathematici
PY - 1996
VL - 63
IS - 2
SP - 103
EP - 113
AB - We derive and investigate integral inequalities of Opial type: $∫_I s|hḣ|dt ≤ ∫_I rḣ² dt$, where h ∈ H, I = (α,β) is any interval on the real line, H is a class of absolutely continuous functions h satisfying h(α) = 0 or h(β) = 0. Our method is a generalization of the method of [3]-[5]. Given the function r we determine the class of functions s for which quadratic integral inequalities of Opial type hold. Such classes have hitherto been described as the classes of solutions of a certain differential equation. In this paper a wider class of functions s is given which is the set of solutions of a certain differential inequality. This class is determined directly and some new inequalities are found.
LA - eng
KW - absolutely continuous function; integral inequality; absolutely continuous functions; quadratic integral inequalities of Opial type; differential inequality
UR - http://eudml.org/doc/262697
ER -

References

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  1. [1] P. R. Beesack, On an integral inequality of Z. Opial, Trans. Amer. Math. Soc. 104 (1962), 470-475. Zbl0122.30102
  2. [2] D. W. Boyd, Best constants in inequalities related to Opial's inequality, J. Math. Anal. Appl. 25 (1969), 378-387. Zbl0176.12702
  3. [3] B. Florkiewicz, Some integral inequalities of Hardy type, Colloq. Math. 43 (1980), 321-330. Zbl0473.26007
  4. [4] B. Florkiewicz, On some integral inequalities of Opial type, to appear. 
  5. [5] B. Florkiewicz and A. Rybarski, Some integral inequalities of Sturm-Liouville type, Colloq. Math. 36 (1976), 127-141. Zbl0354.26007
  6. [6] D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer, Dordrecht, 1991, 114-142. Zbl0744.26011
  7. [7] C. Olech, A simple proof of a certain result of Z. Opial, Ann. Polon. Math. 8 (1960), 61-63. Zbl0089.27404
  8. [8] Z. Opial, Sur une inégalité, Ann. Polon. Math. 8 (1960), 29-32. Zbl0089.27403
  9. [9] R. Redheffer, Inequalities with three functions, J. Math. Anal. Appl. 16 (1966), 219-242. Zbl0144.30702
  10. [10] R. Redheffer, Integral inequalities with boundary terms, in: Inequalities II, Proc. II Symposium on Inequalities, Colorado (USA), August 14-22, 1967, O. Shisha (ed.), New York and London 1970, 261-291. 
  11. [11] G. S. Yang, On a certain result of Z. Opial, Proc. Japan Acad. 42 (1966), 78-83. Zbl0151.05202

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