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MIT PRIMES/Art of Problem Solving

CROWDMATH 2016: Pattern Avoidance

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Functional Reciprocal Theory
adityaguharoy   18
N Aug 16, 2018 by alexanderoldspartan
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 $f: \mathbb{R^+} \to \mathbb{R^+}$ 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


Can we extend this to $3$ variables ?
If yes, how ?
Please help !!
18 replies
adityaguharoy
Nov 21, 2016
alexanderoldspartan
Aug 16, 2018
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