In mathematics, Hardy's theorem is a result in complex analysis describing the behavior of holomorphic functions. Let be a holomorphic function on the open ball centered at zero and radius in the complex plane, and assume that is not a constant function. If one defines for then this function is strictly increasing and is a convex function of . See also Maximum principle Hadamard three-circle theorem References John B. Conway. (1978) Functions of One Complex Variable I. Springer-Verlag, New York, New York. Theorems in complex analysis