Cauchy Transform and roots of polynomials

Question

I am reading this paper by Marcus,Spielman,Srivastava that study finite convolutions of polynomials. I have a question regarding section 1.2 of the paper.

Let $p$ be a real rooted polynomial with roots $\lambda_1,\cdots,\lambda_d$. Define the Cauchy transform as

$G_p(x) = G_\lambda(x) = \frac{1}{d}\sum_i\frac{1}{x-\lambda_i}$

And the inverse Cauchy transform

$K_p(w)=\max\{x: G_p(x)=w\}$

Then show that $K_p(w)$ is the value of $x$ which is larger than all the $\lambda_i$ for which $G_p(x)=w$.

Answer 1

You need to show $\mbox{argmax}\{x:G_p(x)=w\}>\lambda_i$ for all $w,i$. $G_p(x)$ blows up to either $\pm\infty$ at each $\lambda_i$, and tends to zero from above for $x\rightarrow\infty$, so by the intermediate value theorem, the argmax must be bigger than all the roots.