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Delange's characterization of the sine function

Published online by Cambridge University Press:  18 May 2009

Chin-Hung Ching  and
Charles K. Chui
Chin-Hung Ching
Affiliation:
University of Melbourne, Parkville, Victoria, Australia, 3052
Charles K. Chui
Affiliation:
Texas A and M University, College Station, Texas 77801
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In [2], H. Delange gives the following characterization of the sine function.

Theorem A. f(x)=sin x is the only infinitely differentiable real-valued function on the real line such that f'(O)= 1 and

for all real x and n = 0,1,2,….

It is clear that, if f satisfies (1), then the analytic continuation of f is an entire function satisfying

for all z in the complex plane. Hence f is of at most order one and type one. In this note, we prove the following theorem.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 15 , Issue 1 , March 1974 , pp. 66 - 68
Copyright
Copyright © Glasgow Mathematical Journal Trust 1974

References

REFERENCES

1.Boas, R. P., Entire Functions (New York, 1954).Google Scholar
2.Delange, H., Caractérisations des fonctions circulaires, Bull. Sc. Math. 91 (1967), 65–73.Google Scholar
3.Duffin, R. J. and Schaeffer, A. C., On the extension of a functional inequality of S. Bernstein to non-analytic functions, Bull. Amer. Math. Soc. 46 (1940), 356–363.CrossRefGoogle Scholar