Question

Let x, y, z be integers, not all simultaneously equal. If ω is a cube root of unity with Im(ω)≠1, and if f(z)=az^2+bz+c, then find the range of |f(ω)|.

(a) (0, ∞)

(b) [1, ∞)

(c) (√3/2, ∞)

(d) [1/2, ∞)

This question was addressed to me in an online quiz.

The query is from Functions of a Complex Variable topic in division Complex Function Theory of Engineering Mathematics

Answer 1

Correct answer is (b) [1, ∞)

To explain I would say: ω=-1/2+i√3/2. Therefore, |f(ω)|=|a+b(-1/2+i√3/2)+c(-1/2-i√3/2)|

=|(2a-b-c)/2+i(b√3-c√3)/2|=1/2[(2a-b-c)^2+3(b-c)^2]^1/2={1/2[(a-b)^2+(b-c)^2+(c-a)^2]}^1/2. Putting b=c=0 and a=1 gives us the minimum value=1, while, a=∞ gives us the maximum value=∞.