$Let S=S_1\cap S_2\cap S_3 where$

$S_1=\left\{z|z\in C,|z|<4\right\}$

$S_2=\left\{z|z\in C,I_m\left(\frac{z-1+i\sqrt{3}}{1-i\sqrt{3}}\right)>0\right\}$

$S_3=\left\{z|z\in C,R\left(z\right)>0\right\}$

Then the area of S is

Let  $z=x+iy$
$S_1$ is interior of circle of radius 4 units
$S_3$  denotes, x > 0
 $$\begin{gathered} S_2: I_m\frac{(x-1)+i(y+\sqrt{3})}{1+3}\times (1+i\sqrt{3})>0 \\ {S}_2: [\sqrt{3}(x-1)+(y+\sqrt{3})]>0 \end{gathered}$$

$y+\sqrt{3}x>0$
Now shaded region represents the required area 
 $\therefore$ Required area  $=\frac{\text{Area\hspace{0.17em}of\hspace{0.17em}circle}}{4}+\text{Area\hspace{0.17em}of\hspace{0.17em}sector}$

$=\frac{\pi {\left(4\right)}^2}{4}+\frac{2}{3}\frac{\pi {\left(4\right)}^2}{4}=4\pi +\frac{8\pi }{3}=\frac{20\pi }{3}$