Boundary behaviour of holomorphic functions of bounded type (Q2756223)

Let \(H^\infty(\mathbb{D}) =\{f:f\) is bounded and holomorphic in the unit disk \(\mathbb{D}\}\) and \(N=\{f-g_1/g_2 :g_1\) and \(g_2H^\infty (\mathbb{D})\}\) (Nevanlinna class). It is known that \(\forall f\in N\), \(3f_*(e^{it})= \lim_{r\to 1}f (re^{it})\) a.e. for \(e^{it}\) \(\varepsilon\) (the unit circle) and \(\ln |f_*|\in L^1(\mathbb{T})\). The positive sequence \(\sigma(u))_{n\geq 0}\) verifies the condition (C) if it is log-concave and tends to \(+\infty\), i.e., \(\sigma(n)^2 \geq\sigma(n-1) \sigma(n+1))\) and \(\sigma(n)\to+ \infty (n\to+ \infty)\). It verifies the condition (R) ifNEWLINENEWLINENEWLINEa) \(\sigma(0)=1\), \((\sigma(n))_n\) is log-concave and tends to \(+\infty\);NEWLINENEWLINENEWLINEb) for a constant \(\gamma\in(0,1/2)\) and \(n\geq 1\), \((\ln\sigma(n) /n^\gamma) \downarrow\);NEWLINENEWLINENEWLINEc) \(\sup\{\sigma(n)/ \sigma(n+1): n\geq 0\}=1.\)NEWLINENEWLINENEWLINEIt verifies the condition (S) ifNEWLINENEWLINENEWLINEa') \(\sigma (0)=0\), \((\sigma(n)^2)_n\) is concave and tends to \(+\infty\); b') for \(n\geq 0\), \((\ln\sigma(n)/ \ln(n+1)) \downarrow 0\);NEWLINENEWLINENEWLINEc') the same as in c) above.NEWLINENEWLINENEWLINELet \((\alpha(n))_{n\geq 0}\) and \((\beta(n))_{n\geq 0}\) verify the condition (C) and let \({\mathcal N}_{\alpha,\beta} =\{f(z)=\sum_{n\geq 0}a_nz^n\in N:f\in L^2 (\mathbb{T})\), \(\sum_{n\geq 0}(|a_n|^2/ \beta(n)^2)< +\infty\) and \(\sum_{n\geq 1}|\widehat f_*(-n)^2 |^2\alpha(n)^2 <+\infty\}\). The author proves: NEWLINENEWLINENEWLINE1) Suppose that \((\alpha(n))_n\) and \((\beta(n))_n\) verify the condition (R) and \(\lim_{n\to+\infty} (\ln\alpha(n)/ \ln n)>0\) and \(\sup\{\ln\alpha (n^2)/\ln \alpha(n): n\geq 1\}< +\infty\). Then \({\mathcal N}_{\alpha, \beta}=H^2 (D) \Leftrightarrow \sum_{n\geq 1}[(\ln\alpha (n+1)-\ln\alpha(n))/ \ln\beta(n)]= + \infty\).NEWLINENEWLINENEWLINE2) Suppose that \(\alpha(n)= \exp(cn^\tau)\), where \(\tau\in (0,1/2)\) and \(c\) is a positive constant, and that \((\beta(n))_n\) verifies the condition (R). Then NEWLINE\[NEWLINE\sum_{n\geq 1}\bigl(n^{(1-2 \tau)/(1- \tau)}(\ln\beta (n)\bigr)^{1/ (1- \tau)})^{-1}= +\infty \Rightarrow{\mathcal N}_{\alpha, \beta}=H^2 (\mathbb{D})NEWLINE\]NEWLINE and \(\forall\varepsilon >0\) such that NEWLINE\[NEWLINE\sum_{n\geq 1}\bigl(\ln n)^{(\tau/(1-\tau) \bigr) +\varepsilon} \Bigl(n^{1-2\tau \over 1-\varepsilon} \bigl(\ln\beta (n) \bigr)^{1\over 1-\tau} \Bigr)^{-1}< +\infty\Rightarrow H^2(D) \subsetneqq{\mathcal N}_{\alpha, \beta}.NEWLINE\]NEWLINE 3) Suppose that \((\alpha(n))_n\) and \((\beta(n))_n\) verify the condition (S). Then \(\beta(n)=O (\alpha(n)^p)\) for some \(p>0\Rightarrow {\mathcal N}_{\alpha, \beta}=H^2 (\mathbb{D})\); \(\sum_{n\geq 1}[(\ln\alpha (n+1)-\ln \alpha( n))/ \ln\beta(n)] <+\infty \Rightarrow H^2(\mathbb{D}) \subsetneqq{\mathcal N}_{\alpha,\beta}\).