Jensen type inequalities and radial null sets (Q2718033)

Let \(k\) denote an increasing smooth non-negative and integrable on [0,1) function satisfying the condition NEWLINE\[NEWLINEk(1-\tfrac{t}{2})\leq Ck(1-t),\quad 0<t<\tfrac 12).\tag{1}NEWLINE\]NEWLINE Let \(A^{\langle k\rangle}\) denote the Banach space of analytic functions in the unit disk \(D\) with the norm \(\|f\|_{\langle k\rangle}=\sup\{|f(z)|\exp(-k|z|):\;z\in D\}<\infty\). Extending the Jensen formula the author gives an upper estimate of \(\log |f(0)|\) in terms of a finite set \(E\in\partial D\). Let \(J(E,\varphi,k)=\sup\{\log|f(0)|:\;f\in UBA^{\langle k\rangle}\cap C(D\cup E)\), \(|f||_E=\varphi \}\), where \(E\) is a closed set, \(\varphi\geq 0\) and \(UBA^{\langle k\rangle}\) denotes the unit ball of \(A^{ \langle k\rangle}\). The main result contains the following Theorem. NEWLINE\[NEWLINEJ(E,\varphi,k)\leq\int_E \max\{\log\varphi (\zeta), \log p\} dm(\zeta)-(\log p)\frac{\alpha}{1-\alpha}+ \left(\frac{L}{\alpha}\right)^{\log_2 C}Entr_k(E)NEWLINE\]NEWLINE where \(0<p\leq 1, 0<\alpha\leq \frac 12\) are arbitrary, \(C\) is constant in (1), \(L\) is an absolute constant and \(Entr_k(E)=\sum_n \int_{1-|I_n|}^1 k(t) dt\) where \(\{I_n\}\) are the complementary arcs of \(E\). As application the study of radial null sets is also considered.