Norm inequalities of exponential type for holomorphic functions

Abstract

Let Δ_ρ={z∈C : |z| <ρ} with ρ=1, ∞ and let H(Δ_ρ) stand for the class of holomorphic functions in Δ_ρ. Let φ∈H(Δ_ρ) with Δ_ρ being the domain of holomorphy of φ and φ(0)=0, φ⁽ⁿ⁾(0)>0 for n=1, 2, …. Then k(z, ζ)≡φ(z\bar{ζ}) is the reproducing kernel of a uniquely determined Hilbert space H_φ of functions f∈H(Δ_ρ) with f(0)=0 and norm ||f||_φ. The function ψ≡expφ also determines a unique Hilbert space H_ψ of functions g∈H(Δ_ρ) with norm ||g||_ψ and such that K(z, ζ)≡ψ(z\bar{ζ}), z, ζ∈Δ_ρ, is its reproducing kernel. The following is proved: Let f∈H_φ, then exp f∈H_ψ and
||exp f||_ψ²{≤}exp ||f||_φ²
with equality if and only if f is of the form f(z)=k(z, ζ)=φ(z\bar{ζ}) for some ζ∈Δ_ρ. The method of proof of this sharp inequality is based on ideas of both Aronszajn and Milin, and it can be extended by replacing the exponential function by any entire function with non-negative Taylor-coefficients. We also give several applications of this inequality in the theory of entire functions and functions holomorphic in the unit disk.