Final answer:
An example of a non-constant bounded holomorphic function on an unbounded open subset of ℂ is given by the function f(z) = e^z, where z is a complex number.
Step-by-step explanation:
An example of a non-constant bounded holomorphic function on an unbounded open subset of ℂ is given by the function f(z) = e^z, where z is a complex number. The exponential function is holomorphic everywhere in the complex plane, including unbounded open subsets.
To show that f(z) is non-constant, we can consider the argument of f(z). Since e^z can take any value on the complex plane, the argument of f(z) can also take any value. This means that f(z) cannot be constant.
Furthermore, we can show that f(z) is bounded by considering the modulus of f(z). Since e^z is defined as e^x * e^iy, where x and y are the real and imaginary parts of z, the modulus of f(z), |f(z)|, is equal to |e^x * e^iy|. Since both e^x and e^iy are bounded functions, their product e^x * e^iy is also bounded. Therefore, f(z) = e^z is a non-constant bounded holomorphic function on an unbounded open subset of ℂ.