Mobius transformation :
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     Substituting the values of $z_{1}$, $z_{2}$, $z_{3}$, $w_{1}$, $w_{2}$ and $w_{3}$, we get MATH Therefore, the required Mobius transformation is MATH.
Observations: The points $z_{1} = 0$, $z_{2} = 1$, $z_{3} = \infty$ lie on the line $L$ (say), namely, the real axis of the $z$-plane. According to the orientation induced by these points $z_{1} = 0$, $z_{2} = 1$, $z_{3} = \infty$, the left side region of the real axis of the $z$-plane is the upper half plane MATH. Now, the points $w_{1} = 1$, $w_{2} = i$, $w_{3} = -1$ lie on the circle $C$ (say) centered at the origin and radius $1$ (that is, unit circle $\vert w \vert = 1$) with the counterclockwise orientation in the $w$-plane. The left side region of the unit circle $C$ is $\vert w \vert < 1$ which is the interior of the unit cicle $\vert w \vert = 1$ in the $w$-plane. Since the Mobius transformation maps the left region of $L$ onto the left region of $C$, the above computed Mobius transformation MATH maps the upper half plane MATH onto the region $\vert w \vert < 1$.
 
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