(a) If f(z) is not a constant and is analytic for all (finite) z, and R and M are any positive real numbers (no matter how large), show that there exist values of z for which |z| > R and |f(z)| > M. Use Liouville’s theorem.

(b) If f(z) is a polynomial of degree n > 0 and M is an arbitrary positive real number (no matter how large), show that there exists a positive real number R such that |f(z)| > M for all |z| > R.

(c) Show that f(z) = e^x has the property characterized in (a) but does not have that characterized in (b).

(d) If f(z) is α polynomial in z, not α constant, then f(z) = 0 for at least one value of z. Prove this. Use (a).