Let be the roots of the polynomial

Prove that:

**Solution**

Suppose the roots of polynomial are where

Let . Then, the are the zeros of in the disk where is chosen such that for .

Jensen’s inequality implies that

Applying Cauchy – Schwartz yields,

Therefore,

Letting and we get the result.

One may also use induction… For example:The 2-norm of a polynomial is defined as

and we have

As before let us write the roots of the polynomial in the order presented above. Consider the polynomial

where . Now,