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  "body": "\n## Statement\n\nLet $S = \\set{z \\in \\C : 0 \\le \\operatorname{Re} z \\le 1}$ be the closed vertical strip.\nLet $f \\colon S \\to \\C$ be continuous and bounded on $S$ and [[holomorphic|0DG0Z72CQD]] on the interior of $S$.\nFor $0 \\le x \\le 1$ let $M(x) = \\sup_{y \\in \\R} \\abs{f(x + iy)}$.\nThen for each $x$ with $0 \\le x \\le 1$,\n$$M(x) \\le M(0)^{1 - x} M(1)^{x}.$$\n\n## Remarks\n\nProof idea: assume $M(0), M(1) \u003e 0$ and put $g(z) = f(z) M(0)^{z - 1} M(1)^{-z}$, so that $\\abs{g} \\le 1$ on both boundary lines.\nFor $\\varepsilon \u003e 0$ the function $g(z) e^{\\varepsilon z^2}$ tends to $0$ as $\\abs{\\operatorname{Im} z} \\to \\infty$ in $S$, since $g$ is bounded.\nThe [[maximum modulus principle|048F2P90WR]] on long rectangles gives $\\abs{g(z)} e^{\\varepsilon (x^2 - y^2)} \\le e^{\\varepsilon}$, and $\\varepsilon \\to 0$ gives $\\abs{g} \\le 1$.\nThe boundedness hypothesis is needed.\nThe function $f(z) = \\exp(i e^{-i\\pi z})$ satisfies $\\abs{f(x + iy)} = \\exp(\\sin(\\pi x) e^{\\pi y})$.\nIt has modulus $1$ on both lines but is unbounded on the line $\\operatorname{Re} z = \\frac{1}{2}$.\nThis theorem is the key step in the Riesz–Thorin interpolation theorem.\n\n## Explanations\n\n### L1 human\n\nLook at a holomorphic function on a vertical strip, and record its largest size on each vertical line.\nThe theorem says that the logarithm of this size is a convex function of the position across the strip.\nSo the size in the middle is controlled by the sizes on the two edges, in a geometric-mean way.\n\n### L2 human\n\nThe theorem is the strip form of the [[Hadamard three-circle theorem|V00Q05ZW1G]] and a special case of the [[Phragmén–Lindelöf principle|FYR6V2NH6C]].\nIts main use is in interpolation: the Riesz–Thorin theorem bounds an operator on $L^{p}$ for intermediate $p$ by applying it to an analytic family of functions.\nThe same idea appears in complex interpolation of Banach spaces and in convexity bounds for the Riemann zeta function in the critical strip.\n",
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