For a complex polynomial $\sum_0^n a_i z^i$ the sum of roots is readily given by the Vieta's formula.  However, after trying to find out the sum of moduli of roots in terms of the coefficients, I feel like it should not be algebraically computable in terms of coefficients.  Then I tried to figure out some bounds for the mean of moduli.  I realised that we can trivially compute a class of bounds by replacing each modulus by some bound for roots like Cauchy's bound.  However, I could not think of some better way.  I have the following questions:<br>
**Q.No1**  Is it possible to find some good bounds for the average of the moduli of roots of a complex polynomial with complex roots with known coefficients?
<br> **Q.No.2**  Now consider a random polynomial of degree $n$ with coefficients distributed as iid standard normal variates.  How can we find the expectation of the mean of moduli of the roots, especially for finite degree polynomials?
<br>Of the many things I was trying, maybe the following is of some interest.
For any polynomial $\sum_0^n a_i z^i,$ we may note that 
$$\text{ |Product of roots| }=\left|\frac{a_0}{a_n} \right|$$
so if $z_{min}$ is the root min minimum modulus, then $$ |z_{min}| \geq \left(\left|\frac{a_0}{a_n} \right| \right)^{1/n} $$ so that 
$$ \frac{\sum_1^n |z_i|}{n} \geq \left(\left|\frac{a_0}{a_n} \right| \right)^{1/n} $$
I don't know, however, how good that bound is and whether we can find some good ones.  I will be obliged for any help/links/suggestions