  Let $G$ be a simple non-compact linear Lie group. Let $\pi$ be any
irreducible unitary representation of $G$ with infinitesimal character
$\Lambda$ whose continuous part is $\nu$. The beautiful Helgason-Jonson bound
in 1969 says that the norm of $\nu$ is upper bounded by the norm of $\rho(G)$,
which stands for the half sum of the positive roots of $G$. The current paper
aims to give a framework to sharpen the Helgason-Johnson bound when $\pi$ is
infinite-dimensional. We have explicit results for exceptional Lie groups.
Ingredients of the proof include Parathasarathy's Dirac operator inequality,
Vogan pencil, and the unitarily small convex hull introduced by Salamanca-Riba
and Vogan.
