the Hadamard three-circles theorem
/ ah-dah-MAR /
The Hadamard three-circles theorem is the rotationally symmetric cousin of the three-lines theorem, set in an annulus (a ring-shaped region between two concentric circles) instead of a strip. The question is the same in spirit: if you know how large a holomorphic function gets on the inner circle and on the outer circle, how large can it get on a circle of intermediate radius? And the answer has the same beautiful convexity: the maximum on the middle circle is controlled, in log scale, by a straight-line interpolation between the inner and outer maxima.
State it precisely. Let f be holomorphic on the closed annulus r_1 <= |z| <= r_3, and let M(r) be the maximum of |f(z)| on the circle |z| = r. The theorem says that M(r) is a logarithmically convex function of log r: that is, log M(r) is a convex function of log r. As an inequality, for r_1 <= r_2 <= r_3, M(r_2)^(log(r_3/r_1)) <= M(r_1)^(log(r_3/r_2)) * M(r_3)^(log(r_2/r_1)). The proof is the three-lines theorem in new coordinates: substitute z = e^w, which turns the annulus r_1 <= |z| <= r_3 into the vertical strip log r_1 <= Re w <= log r_3 and turns circles into vertical lines; the logarithm wraps the rotational symmetry of the annulus into the translation symmetry of the strip, and three-lines does the rest.
Three-circles is the classical tool for studying the growth of entire functions and is the geometric origin of the 'order' and 'type' of an entire function: comparing M(r) on growing circles is exactly how you measure how fast a function grows. It also appears in approximation theory and in bounding power series. An honest caveat to keep the statement straight: the convexity is of log M(r) as a function of log r — a double logarithm in disguise — NOT of M(r) as a function of r. Forgetting the 'log r' on the horizontal axis is the most common mistake. And the function must be holomorphic on the WHOLE closed annulus (no poles inside the ring), or the convexity argument collapses.
For f(z) = z^n the maximum on |z| = r is M(r) = r^n, so log M(r) = n log r — a straight line in log r, hence trivially convex (the boundary case of the theorem, equality throughout). For a genuine inequality, with M(1) = 2 on the inner circle r = 1 and M(4) = 32 on the outer circle r = 4, the bound on the middle circle r = 2 is M(2) <= 2^(log(4/2)/log(4/1)) * 32^(log(2/1)/log(4/1)) = 2^(1/2) * 32^(1/2) = 8.
log M(r) is convex in log r; for z^n it is exactly the straight line n log r.
The convexity is of log M(r) against log r — both axes logarithmic. Plot M against r and you will not see convexity; plot log M against log r and you will. Mislabeling the axes is the classic error.