Construction of a strictly increasing continuous function.

Question

I'm studying Vittorino Pata's article and I have a doubt about the possibility of constructing a strictly increasing continuous function $\mu$ such that $$\mu(0)=0$$ and $$\mu(r) \leq 1- \frac{\rho(r)}{r}, \forall r \in [0,1],$$ where $$\rho: \mathbb{R}^+ \rightarrow [0, \infty)$$ is a continuous increasing function satisfying $$\rho(r)<r, \ \textrm{if} \ \ r>0.$$ (the kind of functions that we have on Boyd-Wong type results).

Thanks a lot for any help.

Best,

Cleto

Answer 1

Most of the structure of $1-\rho(r)/r$ is irrelevant:

If $\psi:(0,1]\to(0,1]$ is continuous there exists a strictly increasing continuous $\mu:[0,1]\to[0,1]$ with $\mu(0)=0$ and $\mu\le\psi$.

Proof: Let $$\mu(r)=r\min_{r\le s\le 1}\psi(s).$$