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

[de1irious]

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!

 

[John Creighto]

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.

 

[mathwonk]

does cauchy's formula help?