TITLE: An estimate involving polynomial of degree 2
QUESTION [8 upvotes]: Let $P(z)=(z-a)(z-b)$ where $a,b$ are any complex numbers such that $|a|\geq 1, |b|\geq 1.$ Then may I know, if
$$\max_{|z|=1}|P'(z)|\leq \left(\frac{1}{2}+\frac{1}{1+|ab|}\right)\max_{|z|=1}|P(z)|$$
is true?

REPLY [3 votes]: It looks like this inequality is not fulfilled for $a=2,b=-1$:
$$\max(P'(e^{ix}))=\max{\left(5-4\cos{x}\right)^{1/2}}=3$$
$$\max(P(e^{ix}))=\max{\left(10+2\cos{x}-8\cos^2{x}\right)^{1/2}}=\frac{9}{2\sqrt{2}}$$
$$3>\left(\frac{1}{2}+\frac{1}{1+2}\right)\frac{9}{2\sqrt{2}}\approx 2.65$$