Question 125
Let $a \in ] 0 ; 1[$ and $n\gt 0$ an integer. we note $f_{n}$ the function defined on $\mathbb{R}$ by $f_{n}(x)=x+\frac{x^{2}}{n}$, for any real $x$. prove that
$$
\frac{a(1-a) n^{2}+2 a^{2} n+a^{3}}{(1-a)^{2} n^{2}+a(2-a) n+a^{2}}\lt \underbrace{\left(f_{n} \circ f_{n} \circ \cdots \circ f_{n}\right)}_{n}(a)\lt \frac{a n+a^{2}}{(1-a) n+a} .
$$
$$
\frac{a(1-a) n^{2}+2 a^{2} n+a^{3}}{(1-a)^{2} n^{2}+a(2-a) n+a^{2}}\lt \underbrace{\left(f_{n} \circ f_{n} \circ \cdots \circ f_{n}\right)}_{n}(a)\lt \frac{a n+a^{2}}{(1-a) n+a} .
$$