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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