Small Entire Functions with Infinite Growth Index

Abstract

In this paper, we prove that given μ>0 there exists a dense linear manifold M of entire functions, such that,[formula]for every f∈M and l straight line and with infinite growth index for all non-null functions of M. Moreover, every non-null function of M has exactly 2([2μ]+1) Julia directions. And if l is a straight line that does not contain a Julia line, then for every f∈M[formula]and for j≥1, f(j) is bounded and integrable with respect to the length measure on l and ∫lf(j)=0.