Nevanlinna counting function and pull-back measure (Q2796725)

The paper recovers a result due to \textit{P. Lefèvre} et al. [Math. Ann. 351, No. 2, 305--326 (2011; Zbl 1228.47028)] which compares two important quantities of the theory of composition operators: the Nevanlinna counting function \(N_\varphi\) and the size of the Carleson window \(S(\xi,h)\) for the measure \(m_\varphi\) (the pull-back measure of the Haar measure \(m\) on the torus induced by \(\varphi\)), equivalently the size of the Carleson box \(W(\xi,h)\).NEWLINENEWLINEActually their main result Theorem 1.1 specifies the behavior of the constants, namely, given \(\varphi:{\mathbb D}\to{\mathbb D}\) analytic, some \(c\in(0,1/8)\), \(\xi\in{\mathbb T}\) and \(h\) small enough, we have NEWLINE NEWLINE\[NEWLINEm_\varphi\big(S(\xi,(1-c)h)\big)\leq\frac{2}{c^2} \sup_{z\in S(\xi,h)\cap{\mathbb D} }N_\varphi(z)NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\sup_{z\in W(\xi,h)\cap{\mathbb D} }N_\varphi(z)\leq\frac{100}{c^2} m_\varphi\big(W(\xi,(1+c)h)\big).NEWLINE\]NEWLINE One of the most interesting points of the paper is that their argument is more elementary than the proof in [Zbl 1228.47028]. The starting point of the proof is the same (Lemma 2.1 is a weak form of the Stanton formula used in [ Zbl 1228.47028]) but their argument for the main result is far shorter and simpler than in [Zbl 1228.47028]: they introduce a rather simple function to conclude.