On inner functions with $H^p$-derivative.

Integrability of the Derivative of a Blaschke Product

“…Ahern and Clark ( [3], Lemma 1, p. 121) found a very practical criterion for membership of the derivative of a Blaschke product in a Hardy space.…”

“…Ahern and Clark [3] considered the condition that the zeros of B converge to 1 nontangentially (which is equivalent to requiring that the zeros belong to a Stolz angle) in Section 4 of op. cit., again in conjunction with a condition on the moduli.…”

“…cit., again in conjunction with a condition on the moduli. By putting together various theorems from different sections of [3], one can readily deduce the following result: if the zeros of B belong to a Stolz angle, then B ∈ ∩ 0<p<1/2 H p and the exponent one-half is sharp. While in our view the statement in this form deserves to be better known, we present it together with a new and very short proof in Section 1.…”

A Reverse Schwarz–Pick Inequality

“…Some special cases of this statement have been known. In particular, it was proved by Ahern and Clark (see [1,Corollary 4]) that, within the current class of θ's, the inner part of θ ′ will be nontrivial provided that θ has a singular factor (because such factors are actually inherited by θ ′ ). On the other hand, if θ is a finite Blaschke product with at least two zeros, then θ ′ is known to have zeros in D (see [11] for more information on the location of these), so it is clear, once again, that θ ′ is non-outer.…”

Composition operators on Dirichlet-type spaces

“…Several authors have studied conditions on inner functions Φ sufficient to imply that Φ ∈ B γ , and thus, Φ ∈ D 2γ [1,2,3,8,18,20].…”