The discussion revolves around a problem involving a polynomial \( p(z) \) of degree \( n \) and the application of the Maximum Modulus Principle to show that \( |p(z)| \leq M |z|^n \) for \( |z| \geq 1 \). Participants are exploring the implications of the principle in the context of analytic functions.
The discussion is active with participants examining the conditions under which the Maximum Modulus Principle applies. There is a recognition that \( \frac{p(z)}{z^n} \) is analytic on the specified domain, and some participants are clarifying the nature of \( M \) as a real number. The conversation reflects a mix of understanding and uncertainty, with no explicit consensus reached yet.
Participants note the lack of specification regarding the variable \( M \) and express curiosity about the implications of the boundary condition \( |z| = 1 \) in relation to the overall problem.
d2j2003 said:I believe f(z) is analytic at infinity if f(1/z) is analytic at z=0
d2j2003 said:so we can say that since p(z)/z^n is analytic at infinity then it is analytic on the domain |z|>1 so according to the maximum modulus principle, p(z)/z^n can have no local maximum on domain (|z|>1) right?
d2j2003 said:and M is just a real number? it is not specified anywhere... and what about |z|=1 since it says on |z|≥1 and we have looked at |z|>1