$H^p$-derivatives of Blaschke products.

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

“…In this paper we extend Linden's theorem [4] when n A N and p > ð1 À aÞ=n where a is in ð0; 1=ðn þ 1ÞÞ. To do this we extend our results obtained in [3] by considering upper bounds to the integral means of the n'th derivative of certain infinite Blaschke products for n > 1.…”

“…In 1976 C. N. Linden [4] gave a generalization to higher order derivatives of a theorem of Protas, which gives a su‰cient condition for the n'th derivative B ðnÞ ðzÞ of an infinite Blaschke product BðzÞ to belong to the class H p for each p in ð0; ð1 À aÞ=n when a is in ð0; 1=ðn þ 1ÞÞ. He also gave some relevant counterexamples to indicate to some extent that his results are the best obtainable for general Blaschke products.…”

Integral means for the first derivative of Blaschke products

“…Similar examples have been considered previously by Linden [3], and it is not di½cult to follow similar applications.…”

On quotients of holomorphic functions in the disc with boundary regularity conditions

“…Every Blaschke product that satisfies (a) has derivatives B(n) E HP for some p = p(n) (see [5]), and then B(n) E N+ . Trivially B extends analitically to T -{1}, yet B cannot be expresed as a quotient f, /f2 with f1 , f2 E A-(B), because E, that verifies (b) and (c), is not the zero set of any function in A°°(D) (see [7]).…”