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Finding a polynomial of degree $\leq 2$ that attains the maximum value for $\frac{1}{|z|^2+1}$
I’m trying to find a polynomial $p(z)$ of degree $\leq 2$ that attains the maximum value for $\frac{1}{|z|^2+1}$ for $z=x+iy \in \mathbb C$.
I’m trying to minimize $p(z)=a+bz+cz^2$ in this way:
$$ \frac{1}{|z|^2+1}=\frac{2(|z|^2-1)}{(|z|^2-1)^2+4}=\frac{2}{(|z|-1)^2+4} $$
then
$$ b=-2, a=\frac{1}{4}, c=0$$
but it doesn’t gives me a function that attains the maximum for $\frac{1}{|z|^2+1}$
so now, I think that if we can find a bound on the $|z|$ we’ll be good with this problem.
We know that
$$ |x^2+y^2|\leq |x|^2+|y|^2 $$
and
$$ |x|^2\leq x^2 $$
so
$$ x^2+y^2\leq \frac{1}{2} $$
so
$$ x^2+y^2 \leq \frac{1}{2} \implies |x|,|y| \leq \frac{1}{\sqrt{2}}$$
if $z \in \mathbb C$ then
$$ \frac{1}{|z|^2+1}\leq \frac{1}{2} \implies |z| \leq
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