  An upper bound of the logarithmic mean is given by a convex combination of
the arithmetic mean and the geometric mean. In addition, a lower bound of the
logarithmic mean is given by a geometric bridge of the arithmetic mean and the
geometric mean. In this paper, we study the bounds of the logarithmic mean. We
give operator inequalities and norm inequalities for the fundamental
inequalities on the logarithmic mean. We give monotonicity of the parameter for
the unitarily invariant norm of the Heron mean, and give its optimality as the
upper bound of the unitarily invariant norm of the logarithmic mean. We study
the ordering of the unitarily invariant norms for the Heron mean, the Heinz
mean, the binomial mean and the Lehmer mean. Finally, we give a new mean
inequality chain as an application of the point-wise inequality.
