ISL 1983

Let $f : [0, 1] \to \mathbb R$ be continuous and satisfy: $$\begin{cases}bf(2x) = f(x), &\text{ if } 0 \leq x \leq 1/2, f(x) = b + (1 - b)f(2x - 1), &\text{ if } 1/2 \leq x \leq 1,\end{cases}$$ where $b = \frac{1+c}{2+c}$ , $c > 0$ . Show that $0 < f(x)-x < c$ for every $x, 0 < x < 1.$

Let $f$ be a real-valued function defined on $I = (0,+\infty)$ and having no zeros on $I$ . Suppose that $$\lim_{x \to +\infty} \frac{f'(x)}{f(x)}=+\infty.$$ For the sequence $u_n = \ln \left| \frac{f(n+1)}{f(n)} \right|$ , prove that $u_n \to +\infty$ as $n \to +\infty.$