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1996 | 63 | 2 | 103-113
Tytuł artykułu

Some quadratic integral inequalities of Opial type

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Rocznik
Tom
63
Numer
2
Strony
103-113
Opis fizyczny
Daty
wydano
1996
otrzymano
1994-06-16
poprawiono
1995-10-16
poprawiono
1995-12-15
Twórcy
  • Institute of Mathematics, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
Bibliografia
  • [1] P. R. Beesack, On an integral inequality of Z. Opial, Trans. Amer. Math. Soc. 104 (1962), 470-475.
  • [2] D. W. Boyd, Best constants in inequalities related to Opial's inequality, J. Math. Anal. Appl. 25 (1969), 378-387.
  • [3] B. Florkiewicz, Some integral inequalities of Hardy type, Colloq. Math. 43 (1980), 321-330.
  • [4] B. Florkiewicz, On some integral inequalities of Opial type, to appear.
  • [5] B. Florkiewicz and A. Rybarski, Some integral inequalities of Sturm-Liouville type, Colloq. Math. 36 (1976), 127-141.
  • [6] D. S. Mitrinović, J. E. Pečarić and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer, Dordrecht, 1991, 114-142.
  • [7] C. Olech, A simple proof of a certain result of Z. Opial, Ann. Polon. Math. 8 (1960), 61-63.
  • [8] Z. Opial, Sur une inégalité, Ann. Polon. Math. 8 (1960), 29-32.
  • [9] R. Redheffer, Inequalities with three functions, J. Math. Anal. Appl. 16 (1966), 219-242.
  • [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] G. S. Yang, On a certain result of Z. Opial, Proc. Japan Acad. 42 (1966), 78-83.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-apmv63z2p103bwm
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