Duren's Problem on Integral Means of the Derivative of a Subordinate Function: the Radius 1/2 for p ≤ 2 and Bounds for p > 2

Let \(g\) be subordinate to \(f\) in the unit disc, that is, \(g=f\circ\varphi\) with \(\varphi\) analytic, \(|\varphi|\lt 1\) and \(\varphi(0)=0\). Goluzin proved in 1951 that \(M_2(r,g')\)\({}\le M_2(r,f')\) for \(r\le 1/2\), and that \(M_p(r,g')\)\({}\le M_p(r,f')\) for every \(p\gt 0\) when \(r\le\sqrt2-1\). Problem 5.39 of Hayman and Lingham's Research Problems in Function Theory, posed by P. L. Duren, asks for the largest number \(r_p\) such that the inequality between the \(p\)-th means of the derivatives holds for \(0\lt r\lt r_p\). We show that \(r_p=1/2\) for every \(0\lt p\le 2\), in the quantitative form \(M_p(r,g')\)\({}\le(\alpha^2\)\({}+4r^2(1\)\({}-\alpha^2))^{1/2}M_p(r,f')\) for \(r\le1/2\), where \(\alpha=|\varphi'(0)|\). The proof is short: Hölder's inequality between Littlewood's subordination theorem and Goluzin's theorem, applied to a power of the zero-free part of \(f'\). For \(p\gt 2\) the problem remains open, and we prove two-sided bounds. The function \(p\mapsto r_p\) is non-increasing and left-continuous, \(r_p\to1/2\) as \(p\downarrow 2\), and \(r_p\)\({}\to\sqrt2\)\({}-1\)\({}=r_\infty\) as \(p\to\infty\). Moreover \(r_p\)\({}\ge r_1(p)\)\({}\gt \sqrt2\)\({}-1\) for every finite \(p\gt 2\), where \(r_1(p)\) is the root in \((\sqrt2-1,1/2)\) of \(8r^4\)\({}-16r^3\)\({}+(p\)\({}+4)r^2\)\({}+2pr\)\({}-p\)\({}=0\); so the lower bound \(\sqrt2-1\) recorded with the problem is not best possible for any finite \(p\). In the other direction, \(r_p\lt 1/2\) for every \(p\ge 12.0068\), with explicit upper bounds for larger \(p\), for instance \(r_{20}\lt 0.4779\) and \(r_{100}\lt 0.4372\); these rest on \(19\) explicit pairs \((f,\varphi)\) violating the inequality, verified in exact rational arithmetic by three independently written programs. The exact value of \(r_p\) for \(p\gt 2\) is not determined; numerical experiments, which prove nothing, suggest that \(r_p=1/2\) up to \(p\approx12.0065\). This is an unrefereed note.