Mean modulus and the fractional derivative of an inner function

Integrability of the Derivative of a Blaschke Product

“…Theorem B can also be deduced from Theorem 6.1 of Ahern's paper [1] and Theorem B by means of the Schwarz-Pick lemma and integration in θ, and the proof works even for arbitrary inner functions. In summary, our contribution here consists in giving an even simpler proof using Proposition 2.…”

“…For a generalization, see Theorem 6.2 of Ahern [1]. Cohn [10] proved the converse of Theorem C for interpolating Blaschke products.…”

“…However, a rather surprising phenomenon occurs here: B ∈ ∩ 0<p<3/2 A p , whenever the zeros of B lie in a Stolz angle. Though this can easily be deduced from Ahern's paper [1], we give a proof based only on the Schwarz lemma and the fact that B ∈ ∩ 0<p<1/2 H p . This is also part of Section 1.…”

“…(1) for Hardy spaces H p by Protas [30], Ahern -Clark [3], Ahern [1], and Cohn [10]; (2) for the related B p spaces, the Banach envelopes of the Hardy spaces with exponents smaller than one, by Ahern and Clark [3], [4]; (3) for weighted Bergman spaces A p α by Ahern [1] and Kim [22]. The statements of many such results can be found in Colwell's monograph [11].…”

The backward shift on Dirichlet-type spaces

“…A variety of results on the Taylor coefficients of inner functions exist in the literature (see [1], for instance) which provide examples of nontrivial invariant subspaces of the form M ϕ ∩ H α .…”