New bounds for a generalized logarithmic mean and Heinz mean

In this paper, by using the monotone form of L’Hospital’s rule and a criterion for the monotonicity of quotient of two power series we present some sharp bounds for a generalized logarithmic mean and Heinz mean by weighted means of harmonic mean, geometric mean, arithmetic mean, two power means \(M_{1/2}(a,b)\) $M_{1/2}(a,b)$ and \(M_{2}(a,b)\) $M_2(a,b)$ . Operator versions of these inequalities are obtained except for those related to the quadratic mean \( M_{2}(a,b)\) $M_2(a,b)$ .