Bounding Zeros of H² Functions via Concentrations

. It is well-known that the zeros fz j g of a function in the classical Hardy space H 2 satisfy P 1 \Gamma jz j j ! 1 ; however, this sum can be arbitrarily large. We shall bound this sum by a constant that depends on the concentration of the function, a concept introduced by Beauzamy and Enflo. 1. INTRODUCTION Consider a function f : D ! C in the classical Hardy space H 2 (D) where D is the open unit disk in the complex plane C. It is well-known that the zeros fz j g of f satisfy P 1 \Gamma jz j j ! 1. However, this sum can be arbitrarily large as seen by considering an appropriate Blaschke product. Fix 1 p 2 and consider the subset A p of H 2 where A p = ff 2 H p (D) : f(z) = X j0 a j z j and X j0 ja j j p ! 1g: Of course, A 2 is just H 2 and A 1 is the usual algebra A+ (T ). For a function in A p with zeros fz j g, we shall bound P 1 \Gamma jz j j by a constant that depends on the concentration of the function, a notation introduced by Beauzamy and Enfl..