Recently there have appeared several works establishing estimates for Riemann-Stieltjes (R-S) integrals. We mention only those of R. Darst and H. Pollard [Proc. Amer. Math. Soc. 25 (1970), 912--913] and of P. R. Beesack [Rocky Mountain J. Math. 5 (1975), 75-78] and refer the reader to the bibliographies therein for further information. Darst and Pollard [op. cit.] studied the integral under classical hypotheses. Their results were generalized by Beesack [op. cit.], who, among other things, established upper and lower bounds for the classical integral as well as for improper integrals. In this paper we attack the problem of determining inequalities for R-S integrals using a generalized variation. In Section 1 we recall the definition of generalized variation and some properties related to it. In Section 2 we discuss briefly the theorems of T. H. Hildebrandt [Introduction to the theory of integration, Academic Press, New York, 1963] related to the R-S integral and we present some further results including a new necessary condition for the existence of this integral. Finally, in Section 3 we determine various estimates for R-S integrals and compare them with one another and with those of Beesack [op. cit.]. Finbally we generalize one estimate to the Weierstrass integral.

Alcune limitazioni per l'integrale di Riemann-Stieltjes e per l'integrale di Weierstrass

CANDELORO, Domenico;PUCCI, Patrizia
1977-01-01

Abstract

Recently there have appeared several works establishing estimates for Riemann-Stieltjes (R-S) integrals. We mention only those of R. Darst and H. Pollard [Proc. Amer. Math. Soc. 25 (1970), 912--913] and of P. R. Beesack [Rocky Mountain J. Math. 5 (1975), 75-78] and refer the reader to the bibliographies therein for further information. Darst and Pollard [op. cit.] studied the integral under classical hypotheses. Their results were generalized by Beesack [op. cit.], who, among other things, established upper and lower bounds for the classical integral as well as for improper integrals. In this paper we attack the problem of determining inequalities for R-S integrals using a generalized variation. In Section 1 we recall the definition of generalized variation and some properties related to it. In Section 2 we discuss briefly the theorems of T. H. Hildebrandt [Introduction to the theory of integration, Academic Press, New York, 1963] related to the R-S integral and we present some further results including a new necessary condition for the existence of this integral. Finally, in Section 3 we determine various estimates for R-S integrals and compare them with one another and with those of Beesack [op. cit.]. Finbally we generalize one estimate to the Weierstrass integral.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11391/117516
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? ND
social impact