Consider the set of polynomial functions $S := \{p \in C[2, 3]: p$ is a non-zero polynomial function on $[2, 3]$ with coefficients in $[-1, 1]$$\}$, is it true that $\inf_{f \in S} \{ \omega(f) + (\max_{x \in [2, 3]} \mid f(x) \mid)^{-1}\}=0$, where $\omega (f) := \max_{x \in [2, 3]} \mid f(x) \mid - \min_{x \in [2, 3]} \mid f(x) \ \mid $.
This problem generates from my attempt of solving the following problem:
Is every continuous function on $C[2,3]$ a uniform limit of polynomial functions on $[2,3]$ with coefficients in $[-1,1]$? (*)
If we proved that the infimum state above is indeed $0$, then we can prove (*) for all monomials, therefore all continuous functions, which give us (*). As a matter of fact, these two problems are easily observed to be equivalent, while the problem in the first paragraph is apparently much more easier to deal with than (*) itself.