The Helgason-Johnson bound in 1969 says that the norm of nu is upper bounded by the norm of rho(G),where nu is the continuous part of the infinitesimal character of any irreducible unitary representation of a real reductive Lie group G,and rho(G) is the half-sum of the positive roots of G. Recently, we find a way to sharpen this bound.In this talk, I will mention the sharpened bound for exceptional Lie groups and its applications.