$\text {let}\ p(z)=a_0+a_1z+\dotsb+a_{n-1}z^{n-1}+z^n,a_i\in\mathbb{C}.$ If $|p(z)|=1 \forall |z|=1$ is it true that $p(z)=z^n$?

Asked Apr 13 '17 at 17:28

Active Apr 13 '17 at 17:38

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$\text {let}\ p(z)=a_0+a_1z+......+a_{n-1}z^{n-1}+z^n,a_i\in\mathbb{C}.$ if $|p(z)|=1 \forall |z|=1$ is it true that $p(z)=z^n$ for n=1,this is true.but is it also for higher value of n ???

edited Apr 13 '17 at 17:38

asked Apr 13 '17 at 17:28

Eklavya

2,671

2

only $a_{n-1}$ or $a_{n-1}z^{n-1}$? Otherwise please specify the pattern of the polynomial. – user300 Apr 13 '17 at 17:35
The answer is yes. See these answers:
[1] https://math.stackexchange.com/questions/3408/characterizing-nonconstant-entire-functions-with-modulus-1-on-the-unit-circle [2] https://math.stackexchange.com/questions/475344/rational-function-with-absolute-value-1-on-unit-circle
– I. J. Kennedy Apr 13 '17 at 17:49

$\text {let}\ p(z)=a_0+a_1z+\dotsb+a_{n-1}z^{n-1}+z^n,a_i\in\mathbb{C}.$ If $|p(z)|=1 \forall |z|=1$ is it true that $p(z)=z^n$?

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