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, lim z→∞ z∈l exp z μ f z = 0 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 lim z→∞ z∈l exp z μ f j z = 0 and for j ≥ 1, f j is bounded and integrable with respect to the length measure on l and l f j = 0