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Two Inequalities*

Published online by Cambridge University Press:  03 November 2016

G.N. Watson
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Extract

(I). The following inequality is a straight generdisation of one of the most important inequalities occurring in elementary analysis. It is consequently of some intrinsic interest, even though it has to do with a determinant.

Let n be a positive integer (≥ 2) and let a, b, . . . , h be n real numbers (unrestricted as to sign) arranged in descending order of magnitude, and no two being equal. Let x be a positive number, which will be regarded as variable.

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Type
Research Article
Information
The Mathematical Gazette , Volume 37 , Issue 322 , December 1953, pp. 244 - 246
Copyright
Copyright © The Mathematical Association 1953

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Footnotes

*

[The Gazette for December 1903 contained a modest mathematical note which described “A method of determining a very rapidly convergent series for the square root of any positive integer H”; the claim put forward for the method was not without warrant, because it was remarked that eight terms of the series thus obtained for √2 yielded its value to 390 places of decimals. This note, from one of the pupils of F. S. Macaulay at St. Paul’s School, was the precursor of a long series of publications in many of the leading mathematical periodicals of the world, establishing their author as a master of all the resources of modern analysis. We are deeply grateful to Professor Watson for allowing us to celebrate his mathematical jubilee by publishing his characteristic communication “Two inequalities” just fifty years to the month after his first appearance as a creative mathematician. Ed.]

References

* [The Gazette for December 1903 contained a modest mathematical note which described “A method of determining a very rapidly convergent series for the square root of any positive integer H”; the claim put forward for the method was not without warrant, because it was remarked that eight terms of the series thus obtained for √2 yielded its value to 390 places of decimals. This note, from one of the pupils of F. S. Macaulay at St. Paul’s School, was the precursor of a long series of publications in many of the leading mathematical periodicals of the world, establishing their author as a master of all the resources of modern analysis. We are deeply grateful to Professor Watson for allowing us to celebrate his mathematical jubilee by publishing his characteristic communication “Two inequalities” just fifty years to the month after his first appearance as a creative mathematician. Ed.]