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Some inequalities for (a + b)p and (a + b)p + (a − b)p

Published online by Cambridge University Press:  23 January 2015

G. J. O. Jameson
G. J. O. Jameson*
Affiliation:
Dept. of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF e-mail: g.jameson@lancaster.ac.uk
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Extract

We start from two simple identities:

For any p > 0 and 0 ≥ b ≥ a, now let

Can we formulate statements about Gp(a, b) that generalise (1) and (2)? We cannot hope for equalities, but perhaps we can establish inequalities which somehow reproduce (1) when p = 1 and (2) when p = 2. For (1), this might mean an inequality of the form Apap ≤ Gp (a, b) ≤ Bpap for certain constants Ap and Bp, and for (2) a similar statement with ap replaced by ap + bp. However, these are not the only possibilities, as we shall see.

Type
Articles
Information
The Mathematical Gazette , Volume 98 , Issue 541 , March 2014 , pp. 96 - 103
Copyright
Copyright © The Mathematical Association 2014

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References

1. Jameson, G. J. O., An approximation to the arithmetic-geometric mean, Math. Gaz. 98 (March 2014) pp. 85–95.Google Scholar
2. Jameson, G. J. O., Inequalities comparing (a + b)p − ap − bp and ap−1 − b + abp−1 , Elemente Math. 68 (2013) pp. 1–6.Google Scholar