Minimum value of $\left|z^2-z+1\right|+\left|z^2+z+1\right|$ for $z\in \mathbb{C}$

Question

(1) If $\left|z\right| = 1$. Then find minimum value of $\left|z^2+z+4\right|$

(2) If $z\in \mathbb{C}.$ Then minimum value of $\left|z^2-z+1\right|+\left|z^2+z+1\right|.$

$\bf{My\; Try}::$ (1) Given $\left|z\right| = 1\Rightarrow z \bar{z} = 1$.

So $\left|z^2+z+4z\bar{z}\right| = |z|\cdot \left|z+4\bar{z}+1\right| = \left|z+4\bar{z}+1\right|$

Now Let $z = x+iy$. Then $\bar{z} = x-iy$ and $|z| =1\Rightarrow x^2+y^2 = 1$

So $\left|x+iy+4x-4iy+1\right| = \left|5x+1-3iy\right| = \sqrt{(5x+1)^2+9y^2}$

So Let $f(x) = \sqrt{(5x+1)^2+9(1-x^2)} = \sqrt{25x^2+1+10x+9-9x^2}$

So $\displaystyle f(x) = \sqrt{16x^2+10x+10}=4\sqrt{x^2+\frac{5}{8}x+\frac{5}{8}}$

So $\displaystyle f(x) = 4\sqrt{\left(x+\frac{5}{16}\right)^2+\left(\frac{5}{8}-\frac{25}{256}\right)}\geq 4\sqrt{\frac{27 \times 5}{256}} = 4\times \frac{3\sqrt{15}}{16} = \frac{3\sqrt{15}}{4}$

which is occur at $\displaystyle x = -\frac{5}{16}$

(2) Now I did not understand how can i solve (II) one

Help Required

Thanks

Answer 1

HINT. The second part is the triangle inequality in reverse. So instead of starting with $|a+b|$ and getting $|a+b|\leq |a|+|b|$. You have $|a|+|b|$ and want to write something like $|a+b|\leq |a|+|b|$.

Answer 2

For part 2:

Let $a = z^2 + 1$ and $b = z$. We have $a - 1 = b^2$. We have \begin{align*} &(|z^2 - z + 1| + |z^2 + z + 1|)^2\\ =\,& |a - b|^2 + |a + b|^2 + 2|a - b|\cdot |a + b| \\ =\,& 2(|a|^2 + |b|^2) + 2|a^2 - b^2|\\ =\,& 2 |a|^2 + 2|a - 1| + 2|a^2 - a + 1|\\ \ge\,& 2|a|^2 + 2\mathrm{Re}(1 - a) + 2\mathrm{Re}(a^2 - a + 1)\\ =\,& 2|a|^2 + 2\mathrm{Re}(a^2) + 4 - 4\mathrm{Re}(a)\\ =\,& 4[\mathrm{Re}(a)]^2 + 4 - 4\mathrm{Re}(a)\\ =\,& [2\mathrm{Re}(a) - 1]^2 + 3\\ \ge\,& 3 \end{align*} where we have used $|u| \ge \mathrm{Re}(u)$, and $|u|^2 + \mathrm{Re}(u^2) = 2[\mathrm{Re}(u)]^2$.

Also, when $z = \frac{1}{\sqrt 2}\mathrm{i}$, we have $|z^2 - z + 1| + |z^2 + z + 1| = \sqrt 3$.

Thus, the minimum of $|z^2 - z + 1| + |z^2 + z + 1|$ is $\sqrt 3$.