Borel transformations on Dirichlet spaces

We study the growth of an entire functionƒ, whose Borel transformγƒ is holomorphic outside a bounded convex regionD _ƒ with boundary curvature bounded away from 0 and ∞. The functionγƒ is assumed to belong to the Dirichlet space, i.e., it satisfies , wheredv(ξ) is the area element. It is shown that forγƒ to satisfy the above conditions, it is necessary and sufficient to have and the growth indicatrixh _ƒ ofƒ satisfies the relation 0<m≤h″(ϕ)+h(ϕ)≤M<∞. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 58–65, July, 1996.