For a positive operator A with 0 < m less than or equal to A less than or equal to M (m, M is an element of R), the Young operator inequality gives as follows: lambdaA + (1 - lambda) greater than or equal to A(lambda) for lambda is an element of [0, 1]. In this note, we prove that the estimation of the converse Young operator inequality is obtained by using Specht's ratio S(t) = t 1/t-1/e log t 1/t-1 and the logarithmic mean L(s, t) = t-s/log t - log s (s, t > 0), that is, we have for a given p under some conditions pA(lambda) + max {L(1, m) log S(m)/p, L(1, M) log S(M)/p} greater than or equal to lambdaA + (1 - lambda) (greater than or equal to A(lambda)) for lambda is an element of [0, 1]. Moreover by using operator means, we consider the converse Young operator inequality related to two operators A and B. Furthermore we discuss reverse inequalities of the Holder-McCarthy inequality and the inequality on the concavity of the logarithmic function.
