Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-14T22:32:35.205Z Has data issue: false hasContentIssue false
  • English
  • Français

A Note on Some Inequalities

Published online by Cambridge University Press:  18 May 2009

T. M. Flett
T. M. Flett
Affiliation:
Department Of Pure Mathematics, The University, Liverpool
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 the course of some recent work on Fourier series [5, 6] I had occasion to use a number of integral inequalities which were generalizations or limiting cases of known results. These inequalities may perhaps have other applications, and it seems worth while to collect them together in a separate note with one or two further results of a similar nature.

For any number k, used as an index (exponent), and such that K > 1, we write k' = k′(k–1), so that k and k′ are conjugate indices in the sense of Hölder's inequality.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 4 , Issue 1 , December 1958 , pp. 7 - 15
Copyright
Copyright © Glasgow Mathematical Journal Trust 1958

References

1.Bliss, G. A., An integral inequality, J. London Math. Soc., 5 (1930), 40–46.CrossRefGoogle Scholar
2.Bosanquet, L. S., The absolute summability (A) of Fourier series, Proc. Edinburgh Math. Soc. (2), 4 (1934), 12–17.CrossRefGoogle Scholar
3.Copson, E. T., An introduction to the theory of functions of a complex variable (Oxford, 1935).Google Scholar
4.Flett, T. M., Some remarks on a maximal theorem of Hardy and Littlewood, Quart. J. of Math. (Oxford 2nd series), 6 (1955), 275–282.Google Scholar
5.Flett, T. M., Some theorems on odd and even functions, Proc. London Math. Soc. (3), 8 (1958), 135–148.CrossRefGoogle Scholar
6.Flett, T. M.. On the absolute summability of a Fourier series and its conjugate series, Proc. London Math. Soc. (3), 8 (1958), 258–311.CrossRefGoogle Scholar
7.Flett, T. M., Some more theorems concerning the absolute summability of Fourier series and power series, Proc. London Math. Soc. (3), 8 (1958), 357–387.CrossRefGoogle Scholar
8.Hardy, G. H., Notes on some points in the integral calculus, LXIV. Further inequalities between integrals, Messenger of Math., 57 (1927–1928), 12–16.Google Scholar
9.Hardy, G. H. and Littlewood, J. E., Notes on the theory of series (XII): On certain inequalities connected with the calculus of variations, J. London Math. Soc., 5 (1930), 34–39.CrossRefGoogle Scholar
10.Hardy, G. H. and Littlewood, J. E., Some properties of fractional integrals. I, Math. Z., 27 (1928), 565–606.CrossRefGoogle Scholar
11.Hardy, G. H. and Littlewood, J. E., A maximal theorem with function-theoretic applications, Acta Math., 54 (1930), 81–116.CrossRefGoogle Scholar
12.Hardy, G. H. and Littlewood, J. E., An inequality, Math. Z., 40 (1935), 1–40.CrossRefGoogle Scholar
13.Hardy, G. H. and Littlewood, J. E., Some more theorems concerning Fourier series and Fourier power series, Duke Math. J., 2 (1936), 354–382.CrossRefGoogle Scholar
14.Hardy, G. H., Littlewood, J. E., and Pólya, G., Inequalities (Cambridge, 1934).Google Scholar
15.Knopp, K.. Über Reihen mit positiven Gliedern (Zweite Mitteilung), J. London Math. Soc., 5 (1930), 13–21.CrossRefGoogle Scholar
16.Titchmarsh, E. C., Additional note on conjugate functions, J. London Math. Soc., 4 (1929), 204–206.CrossRefGoogle Scholar
17.Yano, S., Notes on Fourier analysis (XXIX): An extrapolation theorem, J. Math. Soc. Japan, 3 (1951), 206–305.CrossRefGoogle Scholar
18.Zygmund, A., Some points in the theory of trigonometric and power series, Trans. Amer. Math. Soc., 36 (1934), 586–617.CrossRefGoogle Scholar
19.Zygmund, A., Trigonometrical series (Warsaw, 1935).Google Scholar