抄録
Let Δρ={z∈C : |z| <ρ} with ρ=1, ∞ and let H(Δρ) stand for the class of holomorphic functions in Δρ. Let φ∈H(Δρ) with Δρ 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 Hφ of functions f∈H(Δρ) with f(0)=0 and norm ||f||φ. The function ψ≡expφ also determines a unique Hilbert space Hψ of functions g∈H(Δρ) with norm ||g||ψ and such that K(z, ζ)≡ψ(z\bar{ζ}), z, ζ∈Δρ, is its reproducing kernel. The following is proved: Let f∈Hφ, then exp f∈Hψ 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∈Δρ. 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.
