Sometimes when faced with an inequality, there is an inconvenient condition on the variables that makes everything seem much harder. How do we get rid of the said condition?
In fact, knowing some clever substitutions will often rid those conditions and give us a conditionless inequality, something we all would prefer.
Other times, we are faced with complicated inequality that we're not quite sure how to tackle. In these cases, it's worth it to actually introduce a condition with a substitution.
In either case, clever substitutions will help us immensely.
Given that satisfy , prove that
We apply the substitution to get rid of the condition . The inequality becomes
However, the last inequality is simply Nesbitt's inequality, so we are done.
Given that , prove that
We see that We apply the substitution to get the condition and the inequality as Now notice that the function is convex, so by Jensen's inequality implying that we are done.
Given that are reals satisfying , prove that
Utilizing the substitution with rids the condition and turns the inequality to
where the identity and the well-known inequality were used.
Given that such that , prove that
Using the substitution with rids the condition and reduces the inequality to
It remains to prove
If , prove
We utilize the substitution to get the condition and the inequality reduces to
However, dividing both sides of the condition by gives and substituting this into our inequality, it remains to prove
However, this is true as by AM-GM as , and summing this inequality cyclically gives the inequality we want to prove.
1) Given that are reals satisfying , prove that
2) Given that are reals satisfying , prove that
These two problems, if you didn't notice, are intricately connected: in fact, one is just the other with the substitution of variables .
With this in mind, the substitution for the the condition is and the substitution for the condition is therefore .
Now with this substitution both problems become the inequality
However, by the inequality , so we are done.
Prove that if satisfy the condition , then
Before we solve this problem, some background on the necessary substitution is needed. Given the condition and , we can rid the condition with the substitution such that .
Now back to the problem: we see that if , then clearly the inequality is true. Otherwise, . This means that so the condition simply turns into
Thus, we can use the substitution described above. However, this means that by AM-GM; similarly, so the inequality we seek is true again.