To find a function $f(z)$ that is analytic in a region, bounded throughout the complex plane, but is not a constant function, we can use the following example:

$f(z) = \frac{1}{1+z^2}$

Let's analyze this function:

1. Analytic in a Region:
   • The function $f(z) = \frac{1}{1+z^2}$ is analytic wherever the denominator is not zero.
   • The denominator $1+z^2 = 0$ when $z = \pm i$ (where $i$ is the imaginary unit).
   • Therefore, ( f(z