Hello,
Here I am going to discuss a new theory and a quite useful way of solving inequality problems in which we use the Rearrangement inequality as a tool to develop a new type of inequality the Functional Reciprocal Sum inequality which is described as follows :
This work is authored by Aditya Guha Roy
If for a function over some subset we have
the pairs and are similarly sorted then we say that
is p-monotone over .
And if ,
the pairs and are oppositely sorted then we say that
is i-monotone over .
With this we have some lemmas :
Lemma 1 : (First Functional Reciprocal Sum Inequality (FRS 1)
If is p-monotone over and is i-monotone over then, we have the following inequality :
Use Rearrangement inequality over to establish the inequality and
use Rearrangement inequality over
to establish the inequality
and can be established using Rearrangement inequality or AM-GM inequality. Problems
We can solve many problems using this Lemma:
Problem 1
For all positive reals and all non-negative real numbers we have the following :
Problem 2
For all positive and all nonnegative we have
Problem 3 (CeuAzul)
For all acute angles
Problem 4
For all we have
Can we extend this to variables ?
If yes, how ?
Please help !!