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.