1.3] that $A_\psi(\alpha)\to 1$ as $\alpha \searrow 1$. This proves part (b).

\(c) The proof of part (c) is a straightforward repetition of [@EZ Proposition 8 (c)] with the only difference that instead of the classical Karamata theorem one has to use its generalisation proved in [@GS Proposition 3.2].

\[Thm2\] Let $p\geq 1$, $\psi$ be a function as in Corollary \[equivalencescor\] and $\omega$ an exponentiation invariant extended limit on $L^\infty(0,\infty)$. Assume that $f\in \cap_{q>p} B^{1/q}_{q,q}(S^1)$ is holomorphic. The following quantities are equivalent:

1.  $$\lim_{h\to\tilde{\omega}} \left(\frac{1}{\psi(\mathrm{e}^{\frac{1}{h}})} \int_{\textbf{D}} |f''(z)|^{p+h} (1-|z|^2)^{2p+2h-2} dz\right)^{\frac{p}{p+h}} = \lim_{t\to \omega} \frac1{\psi(t)} \int_0^t F_f(s)^p ds;$$

2.  $$\lim_{h\to\tilde{\omega}} \left(\frac{1}{\psi(\mathrm{e}^{\frac{1}{h}})} \int_{\textbf{T}\times \textbf{N}} |f *W_n)(e^{i\theta})|^{p+h} d\nu(\theta, n)\right)^{\frac{p}{p+h}}  = \lim_{t\to \omega} \frac1{\psi(t)} \int_0^t \Phi_f(s)^p ds;$$

3.  ${\rm Tr}_{\omega,\psi} |H_{\overline{f}}|^p.$\

Here $\tilde{\omega}$ is defined as in Equation .

The proof of this result is a straightforward repetition of that of [@EZ Theorem 2] with the replacement of $\log t$, $1/r$ and the use of [@EZ Proposition 8] by $\psi(t)$, $\psi(\mathrm{e}^{1/h})$ and the use of Proposition \[prop8\], respectively.

Let us place the result Theorem \[Thm2\] in context. Let $B^{1/q+}_{q,q}(S^1)$ denote the subspace of $B^{1/q}_{q,q}(S^1)$ consisting of holomorphic functions. By the results above, we can define two continuous linear mappings $$\begin{aligned}
T_{LP}:B^{1/q+}(S^1)&\to L^q(S^1\times {{\mathbb N}},\nu), \quad 