the three-lines theorem

The three-lines theorem is a precise, quantitative refinement of the maximum principle for a vertical strip. Picture a function defined on the strip between two vertical lines in the complex plane. You know how big it gets on the left edge and on the right edge. The three-lines theorem tells you how big it can get on any vertical line in between — and the answer is a clean, predictable interpolation: the maximum is a 'logarithmically convex' blend of the two edge maxima. It is a maximum principle with a precise interior bound, not just an inequality.

Here is the statement. Let f be holomorphic and bounded on the strip a <= Re z <= b. Let M(x) denote the supremum of |f(z)| along the vertical line Re z = x. The theorem says that log M(x) is a convex function of x. Concretely, for a <= x <= b, M(x) <= M(a)^((b - x)/(b - a)) * M(b)^((x - a)/(b - a)) — the bound at the intermediate line is the weighted geometric mean of the two edge bounds, with weights given by how close x is to each edge. The proof is Phragmén-Lindelöf applied to an auxiliary function f(z) M(a)^((z - b)/(b - a)) M(b)^((a - z)/(b - a)), cleverly chosen so that it is bounded by 1 on both edges; the maximum principle (in its Phragmén-Lindelöf form, since the strip is unbounded) then gives the bound by 1 throughout, which unwinds to the stated inequality.

This theorem is the analytic heart of complex interpolation theory: the Riesz-Thorin interpolation theorem, which lets you deduce that an operator bounded between two pairs of L^p spaces is bounded between all intermediate pairs, is essentially the three-lines theorem in disguise. It is also a standard tool for bounding zeta functions and Dirichlet series in the critical strip. An honest caveat: 'log M(x) is convex' is the precise content — not 'M(x) is convex' (which is false in general). And the boundedness of f on the strip is a genuine hypothesis; drop it and you fall back into the world of Phragmén-Lindelöf, where unchecked growth at infinity can wreck the conclusion.