An upper bound for minimum

Question

For any polynomials of degree $n$ having all its zeros in $|z|\leq K,K\geq 1,$ is it true $\max_{|z|=1}|nP(z)+(a-z)P'(z)|\geq n\min_{|z|=K}|P(z)| $ where $a$ is any complex number with $|a|\geq K?$

Answer 1

As pointed out by Mefitico, the domain on both sides should be $|z|=K$. Consider the function $(nP+(a-z)P')/(nP)$ in the region $|z|\ge K$. The singularity at infinity is removable, so the maximum principle applies. Since the value at $a$ is 1, the maximum modulus on the circle $|z|=K$ must be at least 1. This implies the inequality that is claimed.

Answer 2

Pick $P(z)=z^n$ then:

$$ \max_{|z|=1}|n P(z)+(a-z)P'(z)| = \max_{|z|=1}|n(z^n)+(a-z)n z^{n-1}|=n|a| $$

Then on the other side of the inequality you proposed: $$ n \min_{|z|=K}|P(z)|=1 \min_{|z|=K}|z^n| = |K|^n $$

So we are checking if for any $a$ satisfying $|a|\geq |K| \geq 1$ and any degree $n$: $$ |a| \geq |K|^n $$

The answer is no, sufficing to pick $a=2$, $K=1.5$, $n=2$. Maybe the domain on the left hand side should also be $|z|=K$?