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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
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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

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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.