🚀 go-pugleaf

I am looking for theorems along the following lines:

(a) If p(x) is a dth degree polynomial and |p(x)| < 1 for
1 <= x <= m where m is much larger than d then all the coefficients
of p are very small.

(b) If p(x) is a dth degree polynomial and |p(i)| < 1 for
i = 1,2,...,m where m is much larger than d then all the coefficients
of p are very small.

(c) If p(x) is a dth degree polynomial and |p(i)| < 1 for
i = 1,2,...,m where m is much larger than d then
|p(x)| < 2 for 1 <= x <= m.

I have some results but I suspect that this problem has already been
studied.  I would appreciate any leads.  Thanks.

-- Richard Beigel