the maximum modulus function
Imagine standing at the center of a flat sheet and asking, as you look out along every direction at exactly distance r, what is the largest height the landscape reaches on that circle? For an entire function f — a function holomorphic on the whole plane — the landscape is |f(z)|, the size of f, and the answer to that question, gathered for every radius r, is a single increasing curve M(r). It is the simplest possible ruler for asking 'how big does f get, and how fast?' as you move out toward infinity.
Precisely, M(r) is the maximum of |f(z)| taken over the circle |z| = r: M(r) = max of |f(z)| for |z| = r. By the maximum modulus principle, the largest value of |f| over the whole closed disk |z| <= r is also attained on the boundary circle, so M(r) doubles as the maximum over the entire disk. The function M(r) is non-decreasing in r, and unless f is constant it is strictly increasing and tends to infinity. A small worked picture: for f(z) = e^z we have |e^z| = e^x where x is the real part of z, which is largest when z = r is real and positive, so M(r) = e^r. For a polynomial of degree n, M(r) grows like the constant times r^n.
M(r) is the foundation of growth theory: the order and type of f are defined by comparing M(r) against e^(r^rho). It also obeys a beautiful regularity — log M(r) is a convex function of log r (this is Hadamard's three-circles theorem), so M can never wobble. The one caveat: M(r) measures only the WORST direction, the single direction where |f| is biggest; it says nothing about how f behaves in other directions, where it may be tiny or even zero. Growth measured by M(r) is an upper silhouette, not the full story.
For f(z) = sin z, write sin z = (e^(iz) - e^(-iz)) / (2i). On the imaginary axis z = i y the term e^(-iz) = e^y dominates, so |sin(i y)| = sinh|y| grows like e^|y| / 2. Hence M(r) for sin z grows like e^r / 2 — exponentially — even though sin x stays between -1 and 1 on the real line.
M(r) reveals that the innocent-looking sine is actually an exponentially large function off the real axis.
M(r) only records the single largest value on each circle; an entire function can be enormous in one direction and vanish in another, so M(r) overstates the typical size. It is an upper bound on |f|, not an average.