Small Entire Functions with Infinite Growth Index

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 ∈ Mformula]and for j ≥ 1, f^(j) is bounded and integrable with respect to the length measure on l and ∫_lf^(j) = 0.