We solve the problem of characterizing weights on (0, infinity) for which the inequality involving two possibly different general inner weighted means (integral(infinity)(0)(integral(t)(0)f*(s)(m2)u(2)(s)ds)(p2/m2) w(2)(t)dt)(1/p2) <= C(integral(infinity)(0)(integral(t)(0)f*(s)(m2)u(1)(s)ds)(p1/m1) w(1)(t)dt)(1/p1) holds, where p(1), p(2), m(1), m(2) is an element of (0, infinity) and p(2) > m(2). The proof is based on a new approach combining duality techniques with sharp weighted estimates for iterated integral and supremum operators. (C) 2017 Elsevier Inc. All rights reserved.