There are times when solving inequalities that one has a sum of fractions in which applying the AM-GM inequality to each denominator results in the wrong sign for the resulting expression.
For example (from , p18), if we wish to show that for real numbers with sum that
we may write (equivalent to ), but this implies and so the sign goes the wrong way.
A way around this is to write
Summing this over then gives as desired.
Here are a few more examples demonstrating this technique.
2. (p9 of ) If are positive real numbers with , then
To prove this we write
Next we have as this is equivalent to . This means . Putting everything together,
3. (based on p8 of ) If for and then
By the AM-GM inequality, , so
Summing this over gives
4. (from ) If are positive, then
Once again, focusing on the denominator,
5. (from the 1991 Asian Pacific Maths Olympiad, see  for other solutions) Let be positive numbers with . Then
Here we write
 Zdravko Cvetkovski, Inequalities: Theorems, Techniques and Selected Problems, Springer, 2012.
 Wang and Kadaveru, Advanced Topics in Inequalities, available from http://www.artofproblemsolving.com/community/q1h1060665p4590952
 Cauchy Reverse Technique: https://translate.google.com.au/translate?hl=en&sl=ja&u=http://mathtrain.jp/crt&prev=search