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Let $p(z)=z^{n}+a_{n-1}z^{n-1}+...+a_{0}$ be a polynomial of degree $n\geq1$.

How can you prove that either $p(z)=z^{n}$ or there exists $z'$ with $|z'|=1$ such that $|p(z')|>1$.

Maybe we can use the maximum modulus principle and consider $q(z)=z^{n}p(1/z)$.

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  • $\begingroup$ i think the degree has to be bigger than 1? $\endgroup$ – jorst Mar 2 '14 at 18:25
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    $\begingroup$ Or (to clarify) it doesn't has to, but then: if $n=0$ nothing is to do, if $n=1$ this is just translating the unit circle, so the circle remains itself only if the translation constant is $0$ $\endgroup$ – jorst Mar 2 '14 at 18:47
  • $\begingroup$ I like your last idea. You have $q(0)=1$, so there has to be a point $z'$ on the unit circle for which $|q(z')|\ge1$. What are the cases of equality in the maximum modulus principle? $\endgroup$ – Greg Martin Mar 2 '14 at 22:08
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Let us assume that $\lvert f(z)\rvert\le 1$, for all $\lvert z\rvert=1$, and show that in such case $f$ has to be equal to $z^n$.

Cauchy Integral formula implies that $$ n!=f^{(n)}(0)=\frac{n!}{2\pi i}\int_{|z|=1}\frac{f(z)\,dz}{z^{n+1}}=\frac{n!}{2\pi }\int_0^{2\pi} f(\mathrm{e}^{it})\,\mathrm{e}^{-int}\,dt, $$ and thus, $\int_0^{2\pi} f(\mathrm{e}^{it})\,\mathrm{e}^{-int}\,dt\in\mathbb R\,$ and $$ 1=\frac{1}{2\pi }\int_0^{2\pi} f(\mathrm{e}^{it})\,\mathrm{e}^{-int}\,dt\le \frac{1}{2\pi }\int_0^{2\pi} |f(\mathrm{e}^{it})|\,dt\le 1, \tag{1} $$ since $\lvert\, f(z)\rvert\le 1$, for all $\lvert z\rvert=1$. Therefore, $$ \mathrm{Re}\,\big(f(\mathrm{e}^{it})\, \mathrm{e}^{-int}\big)=|f(\mathrm{e}^{it})|=1, $$ for all $\,t\in[0,2\pi]$, and thus $$\,\mathrm{Im}\,\big(f(\mathrm{e}^{it})\,\mathrm{e}^{-int}\big)=0,$$ for all $\,t\in[0,2\pi]$. Consequently, $$ f(\mathrm{e}^{it})\,\mathrm{e}^{-int}=1,\quad\text{for all $t\in[0,2\pi]$,} $$ or equivalently $$ f(z)=z^n, $$ for $\lvert z\rvert=1$, which implies that $f(z)=z^n$.

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  • $\begingroup$ Thanks for the great answer. Why $|f(z)|\leq 1$ for all $|z|=1$ implies that $|f(e^{it})e^{-int}| =1$? $\endgroup$ – Spock Dec 7 '14 at 9:36
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    $\begingroup$ See my updated answer. $\endgroup$ – Yiorgos S. Smyrlis Dec 8 '14 at 11:11
  • $\begingroup$ How Can we say that (1) and (2) imply f(e^it)e^-int=1 $\endgroup$ – Balasubramannyan S Mar 28 '20 at 6:46
  • $\begingroup$ @BalasubramannyanS Look at the new version of my answer. $\endgroup$ – Yiorgos S. Smyrlis Mar 28 '20 at 11:42

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