Asymptotically sharp bounds in the Hardy-Littlewood inequalities on mean values of analytic functions

Abstract

Let f be analytic in the unit disc, and let it belong to the Hardy space H p, equipped with the usual norm ∥f∥ p. It is known from the work of Hardy and Littlewood that for q > p, the constants C (p,q) :=sup{∫ 1 0(1 - r) -p/q(1/2π ∫ 2π 0 |f(re iθ)| qdθ) p/qdr|∥f∥ p = 1}, with the usual extension to the case where q = ∞, have C(p,q) < ∞. The authors prove that lim q→p( - p/q)C(p,q) = 1, inf p<q≤∞ (1 - p/q)C(p,q) = 1 and max p<q≤∞ (1 - p/q)C(p, q) = C(p, ∞) = π.