(a) Growth of entire functions. 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.
(b) Growth of polynomials. If f(z) is 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) Exponential functions. Show that f(z) = ex has the property characterized in (a) but does not have that characterized in (b).
(d) Fundmental theorem of algebra. If f(z) is a polynomial in z, not a constant, then f(z) = 0 for at least one value of z. Prove this, using (a).