Integral means for the $n$'th derivative of Blaschke products

Proceedings of the St. Petersburg Mathematical Society, Volume XIII

“…Remark: Theorems 2.1 and 3.1 can be easily generalized to obtain estimates for B (k) (z) B (j) (z) as |z| → 1 − . This is a standard technique which can been find for example in [9,11].…”

“…The condition (1.4) has been extensively studied by many authors [1,2,3,9,11,14] to obtain estimates for the integral means of the derivative of Blaschke products.…”

Growth estimates for logarithmic derivatives of Blaschke products and of functions in the Nevanlinna class

“…Note that if fz n g denotes the sequence of all zeros and poles of a function f A N, then X n ð1 À jz n jÞ < y: ð1:3Þ For a A ð0; 1, we will also make use of the more restrictive condition S ¼ X n ð1 À jz n jÞ a < y ð1:4Þ for the zero/pole sequences fz n g. The convergence condition (1.4) is studied, for example, in [1,7,8,9,11,12], which typically deal with the problem of when the derivatives of a Blaschke product can belong to the Hardy spaces H p , and hence to the Nevanlinna class N. See [3] for the basic theory of Hardy spaces. If fz n g is a sequence of nonzero points in D satisfying (1.4) for some a A ð0; 1, then the product…”