  We prove an intrinsic equivalence between strong hypercontractivity and a
strong logarithmic Sobolev inequality for the cone of logarithmically
subharmonic functions. We introduce a new large class of measures, Euclidean
regular and exponential type, in addition to all compactly-supported measures,
for which this equivalence holds. We prove a Sobolev density theorem through
log-subharmonic functions, and use it to prove the equivalence of strong
hypercontractivity and the strong log Sobolev inequality for such
log-subharmonic functions.
