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Revision of Insert constraints into estimates and error terms from Thu, 02/04/2009 - 20:08

Quick description

In analysis and analytic number theory, one sometimes obtains estimates for expressions involving several variables or parameters subject to various constraints concerning their respective sizes. If these estimates have to be carried along and transformed in substantial ways, or if many of them involving different constraints must be combined at some point, it may be useful (if it is possible) to incorporate the constraints inside the estimates by adding terms which make the estimate trivial if the constraints are not satisfied, and reduce to the useful estimate otherwise.

Prerequisites

Basic calculus or even simple combinatorics and sums.

General discussion

The idea of the trick is best explained with concrete elementary examples.

Example 1

Consider the inequality

 \sqrt{x}\leq x.

This is only valid if x\geq 1. However, if we write

 \sqrt{x}\leq x+1

the inequality is valid for x\geq 0, and is not asymptotically worse than the previous one when x gets large.