FRT FRS of degree 2

by adityaguharoy, Dec 12, 2016, 4:40 PM

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

Alternatively one can find something relevant over here.
(Many thanks to Daniel Sitaru)
http://www.ssmrmh.ro/wp-content/uploads/2016/12/FUNCTIONAL-RECIPROCAL-THEORY-DEGREE-2.pdf

If for a function $f: \mathbb{R^+} \to \mathbb{R^+}$ over some subset $S \subseteq \mathbb{R^+}$ we have
$\bullet$ $\forall x \in S , \forall y \in S$ the pairs $(f(x) \cdot x , f(y) \cdot y)$ and $(\frac{f(x)}{x} , \frac{f(y)}{y})$ are similarly sorted then we say that
$f:$ is p-monotone over $S$.
And if ,
$\bullet$ $\forall x \in S , \forall y \in S$ the pairs $(f(x) \cdot x , f(y) \cdot y)$ and $(\frac{f(x)}{x} , \frac{f(y)}{y})$ are oppositely sorted then we say that
$f:$ is i-monotone over $S$.

With this we have some lemmas :

Lemma 1 : (First Functional Reciprocal Sum Inequality (FRS 1)
If $f: \mathbb{R^+} \to \mathbb{R^+}$ is p-monotone over $S \subseteq \mathbb{R^+}$ and $g:\mathbb{R^+}\to \mathbb{R^+}$ is i-monotone over $S$ then, we have the following inequality :
$2 \le \frac{g(x)}{g(y)}+\frac{g(y)}{g(x)} \le \frac{x}{y}+\frac{y}{x} \le \frac{f(x)}{f(y)}+\frac{f(y)}{f(x)}$
Sketch of Proof
Problems
This post has been edited 4 times. Last edited by adityaguharoy, Mar 3, 2017, 3:24 AM
Inequality
FRT
degree 2
FRS degree 2
algebra
function

Comment

2 Comments

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Can we extend this to $3$ variables ?
If yes, how ?

by adityaguharoy, Jan 9, 2017, 3:59 AM

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Hi, can you explain what do you mean by Similarly sorted and oppositely sorted?

by AyanKhan1729, Dec 24, 2022, 6:28 AM

The oldest, shortest words — "yes" and "no" — are those which require the most thought.

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