The Constant of Interpolation in Bloch Type Spaces

Abstract

AbstractIt is known that there exists a constant $$0<\Delta _1 < 1$$ 0 < Δ 1 < 1 such that any $$\Delta _1$$ Δ 1 -separated sequence for the pseudohyperbolic distance in the open unit disk $$\textbf{D}$$ D of $$\mathbb {C}$$ C is interpolating for the classical Bloch space $$\mathcal {B}$$ B . We will prove that $$0.8114< \Delta _1 < 0.9785$$ 0.8114 < Δ 1 < 0.9785 and we will also generalize this result for Bloch type spaces $$\mathcal {B}_{v_p}$$ B v p for $$v_p(z)=(1-|z|^2)^p$$ v p ( z ) = ( 1 - | z | 2 ) p . In particular, we will provide a construction to calculate an estimate of the lower and upper bounds for the corresponding constant of separation $$\Delta _p$$ Δ p for these spaces. We also prove that $$\Delta _p$$ Δ p tends to 1 when $$p \rightarrow \infty $$ p → ∞ .