# Number of solutions

• Apr 10th 2011, 06:51 PM
skyking
Number of solutions
Let $\displaystyle h(z)=\prod_{i=1}^{n}\frac{z-a_i}{1-\bar{a_i} z}$, where $|a_i|<1, i=0,1,2,...,n$. Show that the equation $h(z)=a_0$ has exactly $n$ solutions inside $|z|=1$.

I am trying to use Rouche's theorem here. My initial thought is that the equation above is equivalent to $\displaystyle \prod_{i=1}^{n}(z-a_i)-a_0\prod_{i=1}^{n}(1-\bar{a_i} z)=0$. So we pick $f(z)=\displaystyle \prod_{i=1}^{n}(z-a_i)$ and $\displaystyle g(z)=-a_0\prod_{i=1}^{n}(1-\bar{a_i} z)$. But I am having trouble showing that $|f(z)|>|g(z)|, |z|=1$.

Any help will be appreciated.
SK
• Apr 10th 2011, 07:36 PM
tonio
Quote:

Originally Posted by skyking
Let $\displaystyle h(z)=\prod_{i=1}^{n}\frac{z-a_i}{1-\bar{a_i} z}$, where $|a_i|<1, i=0,1,2,...,n$. Show that the equation $h(z)=a_0$ has exactly $n$ solutions inside $|z|=1$.

I am trying to use Rouche's theorem here. My initial thought is that the equation above is equivalent to $\displaystyle \prod_{i=1}^{n}(z-a_i)-a_0\prod_{i=1}^{n}(1-\bar{a_i} z)=0$. So we pick $f(z)=\displaystyle \prod_{i=1}^{n}(z-a_i)$ and $\displaystyle g(z)=-a_0\prod_{i=1}^{n}(1-\bar{a_i} z)$. But I am having trouble showing that $|f(z)|>|g(z)|, |z|=1$.

Any help will be appreciated.
SK

Put $z=x+iy\,,\,a_k=\alpha_k+i\beta_k$ , and show that $|z-a_k|^2= |1-\overline{a_k}z|^2$

for $|a_k|<1\,,\,|z|=1$

Tonio
• Apr 11th 2011, 05:07 AM
skyking
Quote:

Originally Posted by tonio
Put $z=x+iy\,,\,a_k=\alpha_k+i\beta_k$ , and show that $|z-a_k|^2= |1-\overline{a_k}z|^2$

for $|a_k|<1\,,\,|z|=1$

Tonio

Tonio

Thank you, I was able to show it with your help.

Thanks again

SK