Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-24T16:38:45.961Z Has data issue: false hasContentIssue false
  • English
  • Français

On the minimum modulus of integral functions

Published online by Cambridge University Press:  24 October 2008

M. L. Cartwright
M. L. Cartwright
Affiliation:
Girton College
Get access Rights & Permissions [Opens in a new window]

Extract

1·1. Let f (z) be an integral function of order ρ, and let

It is well known that, if ρ < 1,

This result was first conjectured by Littlewood, who proved it with cos 2πρ in place of cosπρ; it was afterwards proved independently by Wiman and Valiron about the same time. Many other authors have written on the subject, considering among other things the set of valves of r for which m (r) is comparatively large.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 30 , Issue 4 , October 1934 , pp. 412 - 420
Copyright
Copyright © Cambridge Philosophical Society 1934

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

* Littlewood, J. E., Proc. London Math. Soc. (2), 8 (1908), 189–204.CrossRefGoogle Scholar

† Wiman, A., Math. Ann. 76 (1915), 197–211.CrossRefGoogle Scholar

‡ Valiron, G., Annales Fac. Sc. Toulouse, 3 (1913), 117–257CrossRefGoogle Scholar. See also Lectures on the general theory of integral functions (Toulouse) (1923), 125–132Google Scholar, and Opuscula Mathematica A. Wiman dedicata (Lund, 1930).Google Scholar

§ References will be found in the last paper mentioned. See also Whittaker, J. M., Proc. Edinburgh Math. Soc. (2), 2 (1930), 111–128 (115).CrossRefGoogle Scholar

∥ Besicovitch, A. S., Math. Ann. 97 (1927), 677–695.CrossRefGoogle Scholar

* Phragmèn, E. and Lindelöf, E., Acta Math. 31 (1908), 381.CrossRefGoogle Scholar

* Bernstein, V., Annali di Mat. (4), 12 (1933–1934), 173–196 (179)CrossRefGoogle Scholar. This is a generalization of a theorem of Valiron; see Lectures on the general theory of integral functions, 89.

† See Valiron, G., Lectures on the general theory of integral functions, 64–67Google Scholar. It may be observed that we only require a specially simple case. For since all the zeros lie on the negative real axis, M (r) = f (r) and is obviously a regular function of r for all positive values of r.

* Loc. cit. 128–130. See also Bernstein, V., Reale Accad. d'Italia, 1 (1933), 339–401.Google Scholar

† Cf. Cartwright, M. L., Proc. London Math. Soc. (2) (in the press).Google Scholar

* See Cartwright, M. L., Proc. London Math. Soc. (2), 33 (1932), 212Google Scholar. See Theorem V.