## Acta Mathematica

- Acta Math.
- Volume 54 (1930), 81-116.

### A maximal theorem with function-theoretic applications

G. H. Hardy and J. E. Littlewood

**Full-text: Open access**

#### Article information

**Source**

Acta Math., Volume 54 (1930), 81-116.

**Dates**

First available in Project Euclid: 31 January 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.acta/1485887846

**Digital Object Identifier**

doi:10.1007/BF02547518

**Mathematical Reviews number (MathSciNet)**

MR1555303

**Zentralblatt MATH identifier**

56.0264.02

**Rights**

1930 © Almqvist & Wiksells Boktryckeri-A.-B.

#### Citation

Hardy, G. H.; Littlewood, J. E. A maximal theorem with function-theoretic applications. Acta Math. 54 (1930), 81--116. doi:10.1007/BF02547518. https://projecteuclid.org/euclid.acta/1485887846

#### References

- Another proof has since been found by Mr. R. E. A. C. Paley, and will be published in the
*Proceedings of the London Mathematical Society*. - The arguments used in §§ 5–6 are indeed mostly of the type which are intuitive to a student of cricket averages. A batsman's average is increased by his playing an innings greater than his present average; if his average is increased by playing an innings
*x*, it is further increased by playing next an innings*y*>*x*; and so forth. - If the innings to date are 82, 4, 133, 0, 43, 58, 65, 53, 86, 30, the batsman says to himself at any rate my average for my last 8 innings is 58.5′ (a not uncommon psychology).
- Our original proof of this lemma was much less satisfactory; the present one is due in substance to Mr T. W. Chaundy.
- In what follows the symbol ‘Max’, when it refers to an infinite aggregate of values, is always to be interpreted in the sense of upper bound.
- We suppress the straightforward but tiresome details of the proof.
- See for example G. H. Hardy, ‘Note on a theorem of Hilbert’,
*Math. Zeitschrift*, 6 (1919), 314–317, and ‘Notes on some points in the integral calculus’,*Messenger of Math.*, 54 (1925), 150–156; and E. B. Elliott, ‘A simple exposition of some recently proved facts as to convergeney’,*Journal London Math. Soc.*, 1 (1926), 93–96. A considerable number of other proofs have been given by other writers in the*Journal of the London Mathematical Society*. - This would not necessarily be true if the interval were infinite.
- A. Zygmund, ‘Sur les fonctions conjuguées,’
*Fundamenta Math.*, 13 (1929), 284–303. - This very useful inequality is due to W. H. Young, ‘On a certain series of Fourier’,
*Proc. London Math. Soc.*(2), 11 (1913), 357–366. *A*will not occur again in the sense of Section III. Constants*B, C*in future presserve their identity.*Sn*(θ) is formed from the first*n*+1 terms of the Fourier series of*f*(θ), σ*n*(θ) from the first*n*.- When |θ|<σ the maximum is given by
*r*=1, and when |θ|>1/2π by*r*=0. - The usefulness of a kernel of the type of
*X*was first pointed out by Fejér. See L. Fejér, Über die arithmetischen Mittel erster Ordnung der Fourierreihe’,*Göttinger Nachrichten*, 1925, 13–17. - E. Kogbetliantz, ‘Les séries trigonométriques et les séries sphériques’,
*Annales de l'Ecole Normale*(3), 40 (1923), 259–323.Mathematical Reviews (MathSciNet): MR1509252 - There is of course no particular point in the precise shape of
*S*α(θ); it is an area of fixed size and shape including all ‘Stolz-paths’ to*e*^{iθ}inside an angle 2α. The radius vector corresponds to α=0. - J. E. Littlewood, ‘On functions subharmonic in a circle’,
*Journal Lond. Math. Soc.*, 2 (1927), 192–196. - F. Riesz, ‘Über die Randwerte einer analytischen Funktion’,
*Math. Zeitschrift*, 18 (1923), 87–95.

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