Complex Polynomial: Show Coefficients Bounded by Max |p(z)|

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The discussion centers on proving that the coefficients \( a_k \) of the polynomial \( p(z) = a_0 + a_1 z + \ldots + a_n z^n \) are bounded by the maximum modulus \( M \) when \( |z| = 1 \). Participants suggest using derivatives and Cauchy's formula to approach the problem. The substitution \( z = \exp(i \theta) \) is recommended for simplifying the analysis. Ultimately, the goal is to establish a clear relationship between the coefficients and the maximum modulus of the polynomial.

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If p(z)=a0+a1z+...+anz^n, and max|p(z)|=M for |z|=1, show that each coefficient ak is bounded by M.

I'm trying to take derivatives but it's not getting anywhere. Thanks!
 
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de1irious said:
If p(z)=a0+a1z+...+anz^n, and max|p(z)|=M for |z|=1, show that each coefficient ak is bounded by M.

I'm trying to take derivatives but it's not getting anywhere. Thanks!

Did you replace z with:

exp(i \theta)

I suppose differentiating with respect to theata would make sence:

This would give you:

\sum_0^Na_n(-i)^n exp(-i n \theta)=\sum_0^Na_n exp(i n (\theta+\pi/2))=\sum_0^Na_n (i Z)^n

Yeah, I'm stuck to.
 
Last edited:
does cauchy's formula help?
 
Last edited:

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