Two classical properties of the bessel quotient iν+1 /iν and their implications in pde’s

Two elementary and classical results about the Bessel quotient yν =I ν+1 state that on the half-line (0, ∞) one has for ν ≥ −1/2: Iν (i) 0 < yν < 1; (ii) yν is strictly increasing. In this paper we show that (i) and (ii) have some nontrivial and interesting applications to pde’s. As a consequence of them, we establish some sharp new results for a class of degenerate partial differential equations of parabolic type in Rn+1 + × (0, ∞) which arise in connection with the analysis of the fractional heat operator (∂t − Δ)s in Rn × (0, ∞), see Theorems 1.2, 1.4, 1.5 and 1.7 below.