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∘φ with φ analytic, |φ| < 1 and φ(0) = 0. Goluzin proved in 1951 that M_2(r,g′) ≤ M_2(r,f′) for r ≤ 1/2, and that M_p(r,g′) ≤ M_p(r,f′) for every p > 0 when r ≤ √2 − 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 < r < r_p. We show that r_p = 1/2 for every 0 < p ≤ 2, in the quantitative form M_p(r,g′) ≤ (α² + 4r²(1 − α²))^{1/2} M_p(r,f′) for r ≤ 1/2, where α = |φ′(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 > 2 the problem remains open, and we prove two-sided bounds. The function p ↦ r_p is non-increasing and left-continuous, r_p → 1/2 as p ↓ 2, and r_p → √2 − 1 = r_∞ as p → ∞. Moreover r_p ≥ r_1(p) > √2 − 1 for every finite p > 2, where r_1(p) is the root in (√2 − 1, 1/2) of 8r⁴ − 16r³ + (p+4)r² + 2pr − p = 0; so the lower bound √2 − 1 recorded with the problem is not best possible for any finite p. In the other direction, r_p < 1/2 for every p ≥ 12.0068, with explicit upper bounds for larger p, for instance r_20 < 0.4779 and r_100 < 0.4372; these rest on 19 explicit pairs (f, φ) violating the inequality, verified in exact rational arithmetic by three independently written programs. The exact value of r_p for p > 2 is not determined; numerical experiments, which prove nothing, suggest that r_p = 1/2 up to p ≈ 12.0065. This is an unrefereed note.