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Some inequalities for a hypergeometric integral

Published online by Cambridge University Press:  20 January 2009

T. M. Flett
T. M. Flett
Affiliation:
Department of Pure Mathematics, The University, Sheffield
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In various problems in harmonic analysis (4, 5, 6), I have required an estimate for the integral for the hypergeometric function

in the case where a>c−b>0, b>0, and 0 ≦ x < 1 (the integral is then unbounded as x→l−). Although there are innumerable identities for hypergeometric functions, few inequalities for these functions seem to be known, and in estimating the integral (1) I employed ad hoc arguments that made no use of the theory of hypergeometric functions. The estimates obtained were adequate for my purposes, but were far from sharp, and the object of this note is to show that, by assembling a few known facts concerning hypergeometric functions, we can obtain sharp inequalities for the integral (1). We also give (in §4) several related inequalities that can be obtained by transforming the integral.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 18 , Issue 1 , June 1972 , pp. 31 - 34
Copyright
Copyright © Edinburgh Mathematical Society 1972

References

REFERENCES

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