On the Brennan conjecture (Q1392770)

\textit{J. E. Brennan} [J. Lond. Math. Soc. 18, 261-272 (1978; Zbl 0422.30006)] proved that for a hyperbolic simply-connected domain \(\Omega\) on the sphere with \(\Phi\) a conformal mapping of \(\Omega\) onto the unit disc \[ \iint_\Omega|\Phi'|^pdA<\infty\text{ for }{4\over 3}<p<3+\tau \] with an absolute constant \(\tau\). The Brennan conjecture is that \(\tau=1\). \textit{Ch. Pommerenko} [J. Lond. Math. Soc. 32, 254-258 (1985; Zbl 0576.30017)] proved that \(\tau>.399\). \textit{L. Carleson} and \textit{N. G. Makarov} (Ark. Mat. 32, No. 1, 33-62 (1994; Zbl 0805.30018)] gave the following reformulation of the conjecture. Let \(D\) be a simply-connected domain, \(\infty,a_1, \dots, a_m\in \partial D\). Let \(\Phi\) be a conformal mapping of the upper half-plane onto \(D\) satisfying \(\Phi(z)\sim z^2\) as \(z\to\infty\). Suppose the points \(x_j\in R\) are such that \(\Phi(x_j)= a_j\), \(\Phi'(x_j)=0\) and \(\Phi''(x_j)\) exists for \(j=1, \dots,m\). Let \(\beta_j= \beta(D,a_j)= {2\over\Phi''(x_j)}\), \(j=1,\dots, m\). Then \(\sum^m_{j=1} \beta^2_j<1\). They were able only to treat the first step \(m=2\). The authors consider the following example. Let \(\Theta\) be the domain \(C- \{(\infty,0] \cup\bigcup^N_{j=1}[0,a_j]\}\). Let \(a_j=R_je^{i \theta_j}\), \(-\pi< \theta_1<\theta_2<\cdots<\theta_n<\pi\), \(0<R_j<\infty\), \(N>2\). Let \(F_j\), \(j=1, \dots,N\), map \(\Theta\) onto the domain \(H=C-(\infty,0]\) with \(F_j (a_j) =0\), \(\lim_{j\to\infty}| F_j(z)/z|=1\), \(j=1,\dots,N\). Then \(\beta_j =\lim_{z\to a_j}\left| {F_j(z)\over z-a_j} \right|= | F_j'(a_j) |\). Restricting themselves to symmetric domains and using a technique originated by \textit{K. Oikawa} and \textit{J. A. Jenkins} [Ill. J. Math. 15, 664-671 (1971; Zbl 0229.30012)] in a format previously used by \textit{E. Villamor} [Ark. Mat. 33, No. 1, 183-197 (1995; Zbl 0831.30004)] the authors obtain \(\varlimsup_{k\to\infty} | \frac{\sum_{h=-k}^k \beta_r}{\sqrt{n}} |<1\) concluding that these examples will not provide a contradiction to the Brennan conjecture.