Injective isometries in Orlicz spaces

We show that injective isometries in Orlicz space $L_M$ have to preserve disjointness, provided that Orlicz function $M$ satisfies $\Delta_2$-condition, has a continuous second derivative $M''$, satisfies another ``smoothness type'' condition and either $\lim_{t\to0} M''(t) = \infty$ or $M''(0) = 0$ and $M''(t)>0$ for all $t>0$. The fact that surjective isometries of any rearrangement-invariant function space have to preserve disjointness has been determined before. However dropping the assumption of surjectivity invalidates the general method. In this paper we use a differential technique.