Why the maximum principle needs help on an unbounded region
Every theorem in this rung has leaned on one fact: a non-constant holomorphic function cannot have an interior maximum of |f|, so its size is controlled by its boundary values. That is the maximum-modulus principle, and it is exactly what powered the Schwarz lemma two guides ago — bound f on the boundary of the disk, and you have bounded it everywhere inside. The catch is hiding in the word "boundary": the principle in its clean form wants a bounded region, where the boundary really does surround everything.
Move to an unbounded region and the guarantee can collapse. Take the right half-plane Re z > 0 and the function f(z) = e^(e^z). On the boundary, the imaginary axis z = i y, we have e^z = e^(i y), which lies on the unit circle, so |e^z| = 1 and |f(z)| = |e^(e^(i y))| is bounded by a fixed constant. The boundary values are perfectly tame. Yet along the positive real axis f(x) = e^(e^x) explodes faster than any tower you can name. Boundedness on the boundary bought us nothing inside.
The Phragmen-Lindelof idea: outrun the runaway
The repair is beautifully simple. The Phragmen-Lindelof principle says: if f is holomorphic on an unbounded strip or sector, is bounded by M on the boundary, AND does not grow too fast in the interior, then f is bounded by M everywhere — the naive maximum principle is rescued. The phrase "not too fast" is the load-bearing one, and it is calibrated to the shape of the region. The counterexample e^(e^z) above is precisely a function that grows TOO fast, which is why it slips the leash.
The proof technique is a trick worth keeping. You cannot apply the maximum principle to f directly — the region is unbounded. So you multiply f by a tiny taming factor, a function g(z) that is barely below 1 in size but decays just fast enough at infinity to drag f times g down to zero out there. On a large but FINITE piece of the strip, f times g is now genuinely controlled on the whole boundary, the ordinary maximum principle applies, and you read off a bound on f times g. Then let the taming strength go to zero, and the bound you wanted for f survives.
The three-lines theorem: log-convexity across a strip
Now specialise to the most useful shape: the vertical strip a <= Re z <= b. Suppose f is holomorphic and bounded on this strip. For each vertical line Re z = x, write M(x) for the supremum of |f| along that line — the largest size f reaches at horizontal position x. The boundary of the strip is the two outer lines, where the values M(a) and M(b) are known. The question the three-lines theorem answers is: how big can M(x) be at an interior position x between a and b?
The answer is a clean inequality: log M(x) is a CONVEX function of x. Geometrically, if you plot log M against horizontal position, the value at any interior point lies on or below the straight chord joining log M(a) to log M(b). The sup of |f| in the middle cannot bulge above the line you would draw between its two edge values — it is squeezed by a log-linear interpolation. Where the ordinary maximum principle controlled an interior point by the whole boundary, here the interior LINES are controlled by the two outer lines.
Strip: a <= Re z <= b , x = Re z M(x) = sup over the line Re z = x of |f(z)| Three-lines theorem (with a <= x <= b, t = (b - x)/(b - a)): M(x) <= M(a)^t * M(b)^(1 - t) equivalently log M(x) is a CONVEX function of x.
Why log-convexity falls out of the maximum principle
The proof is a single clever multiplier, in the exact spirit of the previous section. Set the constants so that M(a)^t M(b)^(1 - t) is the target bound, and build a function phi(z) = M(a)^((z - b)/(a - b)) M(b)^((z - a)/(b - a)) — an exponential in z chosen so that |phi| equals M(a) on the left edge and M(b) on the right edge, and phi never vanishes. Now look at the quotient g(z) = f(z) / phi(z).
- On the left edge Re z = a, |g| = |f| / M(a) <= M(a) / M(a) = 1, since |f| there never exceeds its own sup M(a). On the right edge the same arithmetic gives |g| <= 1.
- So |g| <= 1 on BOTH boundary lines of the strip — the boundary is now under control, exactly the situation the maximum principle wants.
- Apply Phragmen-Lindelof (g is bounded on the strip, since f is bounded and phi is bounded below) to conclude |g| <= 1 throughout the interior of the strip.
- Unfold the definition: |f| <= |phi| everywhere, and on the line Re z = x this reads |f| <= M(a)^t M(b)^(1 - t). Take the sup over that line to get M(x) <= M(a)^t M(b)^(1 - t).
That last line is precisely log-convexity. The whole argument is the maximum principle wearing a clever change of clothes — divide out an explicit exponential so the two slanted edge constraints become a single flat constraint |g| <= 1, then let Phragmen-Lindelof carry that flatness inward. Notice that we genuinely NEEDED the growth control: without it, the e^(e^z)-style counterexample would let f beat any phi we build, and the inequality would be false.
The three-circles theorem: the same picture, conformally bent
Hadamard's three-circles theorem is the three-lines theorem looked at through a conformal map. Suppose f is holomorphic on an annulus r_1 <= |z| <= r_2, and let M(r) be the maximum of |f| on the circle |z| = r. Then log M(r) is a convex function of log r. Plot the maximum modulus on a log-log scale and the middle circle's value sits on or below the chord between the two edge circles — identical shape to the strip, just in radial coordinates.
The bridge between the two statements is the map w = log z, which is one of our standard conformal maps. It unrolls the annulus r_1 <= |z| <= r_2 into the vertical strip log r_1 <= Re w <= log r_2: a circle of radius r becomes the vertical line Re w = log r. Apply three-lines on the strip, push it back through z = e^w, and "convex in x = Re w" becomes "convex in log r" — the three-circles theorem, for free. The two results are one theorem wearing two coordinate systems.
Step back and see the family. The plain maximum-modulus principle controls one interior POINT by the boundary; the Schwarz lemma controls a self-map of the disk by pinning |f| <= 1 on the boundary circle; three-lines and three-circles control a whole interior LINE or CIRCLE by two outer ones, with log-convexity as the precise shape of the squeeze; and Phragmen-Lindelof is the growth hypothesis that lets all of this survive on unbounded regions. They are one idea — holomorphy makes |f| rigid, and its size is governed from the boundary inward — told at five magnifications.