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On a theorem of S. Bernstein

Published online by Cambridge University Press:  24 October 2008

Shih-Hsun Chang
Affiliation:
Institute for Advanced StudyPrinceton, N.J.

Extract

S. Bernstein (see(l), pp. 198–204) has proved that if

is an entire function of genus zero, then the series

is convergent. By considering the most unfavourable case, when all the (βn) are positive real numbers, and writing

which is convergent by hypothesis, he proved that

.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1952

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References

REFERENCES

(1)Bernstein, S.Leçons sur les propriétés extrémales et la meitteure approximation des fonctions analytiques d'une variable réelle (Paris, 1926).Google Scholar
(2)Borel, E.Leçons sur les fonctions entières (Paris, 1900). 1Google Scholar
(3)Chang, S. H.J. Lond. math. Soc. 22 (1947), 185–9.CrossRefGoogle Scholar
(4)Hadamard, J.J. Math. pures appl. (4), 9 (1893), 171215.Google Scholar
(5)Hardy, G. H., Littlewood, J. E. and Pólya, G.Inequalities (Cambridge, 1934).Google Scholar
(6)Nevanlinna, R.Le théorème de Picard-Borel et la théorie des fonctions méromorphes (Paris, 1929).Google Scholar
(7)Smithies, F.Proc. Lond. math. Soc. (2), 43 (1937), 255–79.Google Scholar
(8)Valiron, G.Bull. Sci. math. (2), 45 (1921), 258–70.Google Scholar
(9)Valiron, G.Lectures on the general theory of integral functions (Toulouse, 1923).Google Scholar