Suppose $p(x)$ is a degree $m$ polynomial whose coefficients are natural numbers. Suppose further that we have $p(1)=n$, $p(2)\leq nm$ and $p(3)=n^2$, and assume that $m\leq \log n$. So $p$ only grows a little from $1$ to $2$ but grows a lot from $2$ to $3$. I am interested as to how well one can estimate $p(x)$ for $x\in(1,2)$.
My feeling is that, because of the positivity of the coefficients, $p$ is convex and $p$ will have to be relatively flat on $(1,2-c)$ for some sufficiently small $c$.
