Small entire functions with extremely fast growth

Abstract

We prove in this note that, given α ∈ (0, 1/2), there exists a linear manifold M of entire functions satisfying that M is dense in the space of all entire functions such that limz→∞ exp(|z|α)f(j)(z) = 0 on any plane strip for every f ∈ M and for every derivation index j. Moreover, the growth index of each nonnull function of M is infinite with respect to any prefixed sequence of nonconstant entire functions.