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

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.