Inverse Hölder Inequalities with Weight ta

In this paper, we establish weighted inverse Hölder inequalities obtained by replacing Lebesgue measure dt by tαdt: ∫10u(t) v(t) tαdt ≥ Cα(p, q)[∫10up(t) tαdt]1/p [∫10vq(t) tαdt]1/q for all nonnegative and concave functions u and v. Here α > −1, 1 < p, q < ∞, 1/p + l/q = 1, and Cα(p, q) = min{ Vα(p, q), Vα(q, p), Wα(p, q)} with the notations [formula] and B denotes the beta function B(p, q) = ∫10tp − 1 (1 − t)q − 1dt. Equality occurs for the choices u(t) = t, v(t) = 1 − t. This settles a problem raised by Barnard and Wells