Hadamard three-hyperballs type theorem and overconvergence of special monogenic simple series

The classical Hadamard three-circles theorem (1896) gives a relation between the maximum absolute values of an analytic function on three concentric circles. More precisely, it asserts that if f is an analytic function in the annulus { z ∈ C : r 1 < | z | < r 2 } , 0 < r 1 < r < r 2 < ∞ , and if M ( r 1 ) , M ( r 2 ) , and M ( r ) are the maxima of f on the three circles corresponding, respectively, to r 1 , r 2 , and r then { M ( r ) } log r 2 r 1 ⩽ { M ( r 1 ) } log r 2 r { M ( r 2 ) } log r r 1 . In this paper we introduce a Hadamardʼs three-hyperballs type theorem in the framework of Clifford analysis. As a concrete application, we obtain an overconvergence property of special monogenic simple series.