We prove an intrinsic equivalence between strong hypercontractivity and a strong logarithmic Sobolev inequality for the cone of logarithmically subharmonic ($\text{LSH}$) 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 $\text{LSH}$ functions and use it to prove the equivalence of strong hypercontractivity and the strong logarithmic Sobolev inequality for such log-subharmonic functions.

We present several functional inequalitiesfor finite difference gradients, such asa Cheeger inequality, Poincaré and (modified) logarithmic Sobolev inequalities,associated deviation estimates,and an exponential integrability property.In the particular case of the geometric distribution on ${\mathbb{N}}$ we use an integration by parts formula to computethe optimal isoperimetric and Poincaré constants,and to obtain an improvement of ourgeneral logarithmic Sobolev inequality.By a limiting procedure we recover the correspondinginequalities for the exponential distribution.These results have applications to interacting spin systems undera geometric reference measure.