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 everyf∈M and for every derivation indexj. Moreover, the growth index of each nonnull function of M is infinite with respect to any prefixed sequence of nonconstant entire functions.