Question

Let f(z)=|z|^2+Re z(2(z+z̅)+3(z-z̅)/2i, the find the maximum value of |z|^2/f(z).

(a) 1

(b) 2

(c) 3

(d) 4

I had been asked this question during an online interview.

Query is from Functions of a Complex Variable topic in section Complex Function Theory of Engineering Mathematics

Answer 1

Right choice is (b) 2

To explain: Write z=|z|(cosθ+isinθ) ⇒ |z|^2/f(z)=1/(1+4cos^2θ+3sinθcosθ)

=1/(1+4cos^2θ+3/2sin2θ)=1/[2(1+cos2θ)+1+3/2sin2θ].

Now, 2(1+cos2θ)+1+3/2sin2θ=3+2cos2θ+3/2sin2θ≥3-(4+9/4)^1/2=1/2.

Hence, maximum value is 2.