  We prove strong hypercontractivity (SHC) inequalities for logarithmically
subharmonic functions on $\RR^n$ and different classes of measures: Gaussian
measures on $\RR^n$, symmetric Bernoulli and symmetric uniform probability
measures on $\RR$, as well as their convolutions. Surprisingly, a slightly
weaker strong hypercontractivity property holds for {\em any} symmetric measure
on $\RR$. For all measures on $\R$ for which we know the (SHC) holds, we prove
that a log--Sobolev inequality holds in the log-subharmonic category with a
constant {\em smaller} than the one for Gaussian measure in the classical
context. This result is extended to all dimensions for compactly-supported
measures.
