If $\;f\;:\;[a,b]\;\rightarrow \;\mathbb{R}\;$ is monotonic, then
$$\int_a^b \left | f(x)-f \left (\frac{a+b}{2} \right ) \right | dx \;\leqslant \;\int_a^b \left | f(x)-c \right | dx,$$
whatever the number $c \in \mathbb{R}.$
Let $f\;:\;(0,a)\;\rightarrow\;\mathbb{R} \;\;(a>0)$ be a continuous function
such that $0<f(x)<x$ for $x \in (0,a)$. We define by recurrence
$$f_1(x)=f(x),\;f_{n+1}(x)=f(f_n(x)),\;n=1,2,\dots$$
If for some number $m>0$ there exist
$$\lim_{x \to 0+}\;[ ( f(x))^{-m}-x^{-m}]=p>0,$$
then for any $x \in(0,a)$ we have
$$\lim_{n \to \infty}\;(n\cdot p)^{m^{-1}} \cdot f_n(x) =1.$$
3.[]
Let
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