Suppose I have a polynomial
$$
p(x)=\sum\_{i=0}^n p\_ix^i.
$$
For simplicity furthermore assume $p\_n=1$. 


As it is well known we may use Gershgorin circles to give an upper bound for the absolute values of the roots of $p(x)$. The theorem states that all roots are contained within a circle with radius
$$
r=\max\{|p\_0|, 1+|p\_1|,\ldots, 1+|p\_{n-1}|\}.
$$




---


Now I wonder if there is something like an inverse of this theorem. Suppose I know that all roots are contained within a circle of radius $r$. Is there anything that can be said about the maximum coefficient, i.e.
$$
\max |p\_i|\leq \text{some function of }r?
$$


I would also be graceful for a counter example.



Estimate maximal coefficient of a polynomial from a circle containing all roots
There are actually stronger estimates that deal with all of the roots. Let $r\_1,\ldots,r\_n$ be the roots of your polynomial (with multiplicities as appropriate). Then 
$$
 \max\_{0\le i\le n} |p\_i| \ge \frac{1}{4^n} \prod\_{j=1}^n \max\bigl\{|r\_j|,1\bigr\}
$$
and
$$
 \max\_{0\le i\le n} |p\_i| \le 4^n \prod\_{j=1}^n \max\bigl\{|r\_j|,1\bigr\}.
$$
For references to much more general results, see the answers to [Bounds on coefficients of factors of a multivariate polynomial](https://mathoverflow.net/questions/151556/bounds-on-coefficients-of-factors-of-a-multivariate-polynomial/151564#151564) and [Getting a bound on the coefficients of the factor polynomial](https://mathoverflow.net/questions/108726/getting-a-bound-on-the-coefficients-of-the-factor-polynomial?rq=1) . (The $4^{\pm n}$ constants are not best possible.)


