Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-15T12:11:14.628Z Has data issue: false hasContentIssue false
  • English
  • Français

The SCnP-Integral and the Pn+1-Integral

Published online by Cambridge University Press:  20 November 2018

P. S. Bullen  and
C. M. Lee
P. S. Bullen
Affiliation:
University of British Columbia, Vancouver, British Columbia;
C. M. Lee
Affiliation:
University of Wisconsin-Milwaukee, Milwaukee, Wisconsin
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In [2], we have briefly described, as examples of the general theory developed there, a scale of symmetric Cesaro-Perron integrals, namely SCnP-integral for n = 1, 2, 3, … . The purpose of this paper is to consider the integrals in a greater detail.

As a preliminary, we prove some lemmas, which are also interesting for their own sake, concerning the de la Vallée Poussin derivatives in Section 1, and we also state two deep theorems concerning the n-convex functions in Section 2.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 25 , Issue 6 , December 1973 , pp. 1274 - 1284
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Bullen, P. S., A criterion for n-convexity, Pacific J. Math. 36 (1971), 81–98.Google Scholar
2. Bullen, P. S. and Lee, C. M., On the integrals of Perron type (to appear in Trans. Amer. Math. Soc).Google Scholar
3. Burkill, J. C., The Cesáro-Perron integral, Proc. London Math. Soc. 34 (1932), 314-322. 4# The Cesáro-Perron scale of integration, Proc. London Math. Soc. 39 (1935), 541–552.Google Scholar
5. Burkill, J. C., Integrals and trigonometric series, Proc. London Math. Soc. 1 (1951), 46–57.Google Scholar
6. Cross, G. E., The relation between two definite integrals, Proc. Amer. Math. Soc. 11 (1960), 578–579.Google Scholar
7. Cross, G. E., The relation between two symmetric integrals, Proc. Amer. Math. Soc. 14 (1963), 185–190.Google Scholar
8. Hobson, E. W., The theory of functions of a real variable, 3rd ed. (Cambridge Univ. Press, Cambridge, 1927).Google Scholar
9. James, R. D., Generalized nth primitives, Trans Amer. Math. Soc. 76 (1954), 149–176.Google Scholar
10. James, R. D., Summable trigonometric series, Pacific J. Math. 6 (1956), 99–110.Google Scholar
11. Kuratowski, K., Topology (Academic Press, New York, 1966).Google Scholar
12. Marcinkiewicz, J. and Zygmund, A., On the differentiability of functions and summability of trigonometric series, Fund. Math. 27 (1937), 38–69.Google Scholar
13. Sargent, W. L. C., On the generalized derivatives and Cesâro-Denjoy integrals, Proc. London Math. Soc. 52 (1951), 365–376.Google Scholar
14. Skvorcov, V. A., Concerning definitions of P2∼ and SCP-integrals, Vestnik Moskov. Univ. Ser. I Mat. Meh. 21 (1966), 12–19.Google Scholar