Orlicz functions that do not satisfy the \Delta_2-condition and high order G\^{a}teaux smoothness in h_M(\Gamma).

We study Orlicz functions that do not satisfy the \Delta _2-condition at zero. We prove that for every Orlicz function M such that \limsup _{t\to 0}M(t)/t^p \!>0 for some p\ge 1, there exists a positive sequence T=(t_k)_{k=1}^\infty tending to zero and such that \begin{equation*} \sup _{k\in \mathbb {N}}\frac {M(ct_k)}{M(t_k)} <\infty,\text { for all }c>1, \end{equation*} that is, M satisfies the \Delta _2 condition with respect to T. Consequently, we show that for each Orlicz function with lower Boyd index \alpha _M < \infty there exists an Orlicz function N such that: (a) there exists a positive sequence T=(t_k)_{k=1}^\infty tending to zero such that N satisfies the \Delta _2 condition with respect to T, and (b) the space h_N is isomorphic to a subspace of h_M generated by one vector. We apply this result to find the maximal possible order of Gâteaux differentiability of a continuous bump function on the Orlicz space h_M(\Gamma) for \Gamma uncountable. [ABSTRACT FROM AUTHOR]