Infimum taken over $\lambda$ in $\mathbb{C}$

Setting one of the summand to minimum, may not minimize the sum

Let $\lambda = x+iy$

$$F=(x-2)^2+y^2+(2x-1)^2+(2y)^2+(x)^2+(y)^2$$

$$=6x^2-8x+5+6y^2$$

$$\ge6x^2-8x+5$$

Again $6x^2-8x+5=6\left(x-\dfrac23\right)^2+5-\left(\dfrac23\right)^2\ge5-6\cdot\left(\dfrac23\right)^2$

So, the minimum value$\left[5-6\cdot\left(\dfrac23\right)^2\right]$ of $F$ occurs if $y=0,x-\dfrac23=0$