MathDB

Let

S\subseteq\mathbb{R}

be a set of real numbers. We say that a pair

(f, g)

of functions from

S

into

S

is a Spanish Couple on

S

, if they satisfy the following conditions: (i) Both functions are strictly increasing, i.e.

f(x) < f(y)

and

g(x) < g(y)

for all

x

y\in S

with

x < y

; (ii) The inequality

f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)

holds for all

x\in S

. Decide whether there exists a Spanish Couple [*] on the set S \equal{} \mathbb{N} of positive integers; [*] on the set S \equal{} \{a \minus{} \frac {1}{b}: a, b\in\mathbb{N}\} Proposed by Hans Zantema, Netherlands

Problem Statement