Hadamard three-circle theorem

For a holomorphic function on an annulus, the logarithm of its maximum modulus on concentric circles is convex as a function of the logarithm of the radius.

Subharmonic function: A function satisfying a local mean-value inequality, generalizing the behavior of harmonic functions. The logarithm of the modulus of a holomorphic function is subharmonic, the source of the theorem's convexity.
Holomorphic function: A complex-valued function that is complex differentiable throughout an open domain. Holomorphy is the essential analytic hypothesis behind the circle-by-circle bound.
Three-circles inequality: The inequality bounding a middle-circle maximum by a geometric interpolation of two surrounding maxima. It is the theorem's direct quantitative form for three chosen radii.
Three-lines theorem: A bound on a holomorphic function in a vertical strip, interpolating its bounds on two boundary lines. It is the strip analogue, with parallel boundary lines replacing concentric circles.
Harmonic function: A twice-differentiable function whose Laplacian vanishes, with values governed locally by their averages. Linear interpolation in log-radius comes from harmonic functions on an annulus.
Annulus: A planar region between two concentric circles, possibly excluding one or both boundary circles. The theorem compares radii lying within a domain of this shape.
Propagation of smallness: A principle by which smallness of a function on one region constrains it on a larger region. The theorem carries small boundary values across intermediate circles.
Hadamard three-spheres theorem: A logarithmic convexity bound for maxima on concentric spheres of a holomorphic or harmonic function. It generalizes the circle geometry to higher-dimensional spheres.
Maximum modulus: The largest absolute value attained by a function on a specified set. The theorem tracks this quantity on each concentric circle.
Convex function: A function whose value at each point of a line segment does not exceed linear interpolation of endpoint values. Convexity applies to the maximum-modulus profile against log-radius.

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