Let \(H(\mathbb C)\) denote the space of all entire functions, endowed with the compact-open topology. For \(f \in H(\mathbb C)\) and \(r>0\) let \(M_f(r):=\max\{|f(z)|: |z|=r\}\). Then the growth of \(m\)-order of \(f\) is defined by \[ \rho_m(f) := \limsup_{r\to\infty}{\frac{\log_{m+1}{M_f(r)}}{\log{r}}} , \] where \(\log_m\) denotes the \(m\)-times iterated logarithm. The growth index of \(f\) is given by \[ i(f):=\min\{\rho_m(f)0\) be given. Then there exists a dense linear manifold \(M\) in \(H(\mathbb C)\) with the following properties: (a) \(\lim_{\substack{ z\to\infty \\ z \in l }} {\exp{(|z|^\mu)}f(z)}=0\) for every \(f \in M\) and any straight line \(l\). (b) \(i(f)=\infty\) for all non-null functions \(f \in M\). (c) Every non-null function \(f \in M\) has exactly \(2([2\mu]+1)\) Julia directions. (d) If \(l\) is a straight line that does not contain a Julia line, then for every \(f \in M\) and \(j \in \mathbb N\) there holds \[ \lim_{\substack{ z\to\infty \\ z \in l }} {\exp{(|z|^\mu)}f^{(j)}(z)}=0. \] In particular, \(f^{(j)}\) is bounded and integrable with respect to the length measure on \(l\) and \(\int_l f^{(j)}=0\). A similar result for \(\mu \in (0,{1 \over 2})\) was proved by \textit{L. Bernal-Gonzalez} [J. Math. Anal. Appl. 207, 541-548 (1997; Zbl 0872.30022)]. The main tool of the proof is a result of \textit{N. U. Arakelyan} and \textit{P. M. Gauthier} [Sov. J. Contemp. Math. Anal., Arm. Acad. Sci. 17, 1-22 (1982; Zbl 0538.30035)] on tangential approximation by entire functions.