Given $b$ and $c$ with $b,c>1$, is it possible to construct a polynomial $p(x)$, whose degree is $n$ for all $c$ and $b$, such that:

• $|p|$ is strictly increasing on $[1,c]$

• and $|b \cdot p(c)| < |p(0)|$?

This might be satisfied by an interpolating polynomial, but how to actually construct it is beyond me.