Improvements of Plachky-Steinebach theorem

We show that the conclusion of Plachky-Steinebach theorem holds true for intervals of the form $\left]\overline{L}_r'(\lambda),y\right[$, where $\overline{L}_r'(\lambda)$ is the right derivative (but not necessarily a derivative) of the generalized log-moment generating function $\overline{L}$ at some $\lambda> 0$ and $y\in\ \left]\overline{L}_r'(\lambda),+\infty\right]$, under the only two following conditions: $(a)$ $\overline{L}'_r(\lambda)$ is a limit point of the set $\left\{\overline{L}'_r(t):t>\lambda\right\}$, $(b)$ $\overline{L}(t_i)$ is a limit for a suitable sequence $(t_i)$. By replacing $\overline{L}_r'(\lambda)$ by $\overline{L}_r'(\lambda^+)$, the above result extends verbatim to the case $\lambda=0$ (replacing $(a)$ by the right continuity of $\overline{L}$ at zero when $\overline{L}_r'(0^+)=-\infty$). No hypothesis is made on $\overline{L}_{]-\infty,\lambda[}$ (e.g. $\overline{L}_{]-\infty,\lambda[}$ may be the constant $+\infty$ when $\lambda=0$); $\lambda\ge 0$ may be a non-differentiability point of $\overline{L}$ and moreover a limit point of non-differentiability points of $\overline{L}$; $\lambda=0$ may be a left and right discontinuity point of $\overline{L}$. The map $\overline{L}_{\mid ]\lambda,\lambda+\varepsilon[}$ may fail to be strictly convex for all $\varepsilon>0$. If we drop the assumption $(b)$, then the same conclusion holds with upper limits in place of limits. The foregoing is valid for general nets $(\mu_\alpha,c_\alpha)$ of Borel probability measures and powers and replacing the intervals $\left]\overline{L}_r'(\lambda^+),y\right[$ by $]x_\alpha,y_\alpha[$ or $[x_\alpha,y_\alpha]$, where $(x_\alpha,y_\alpha)$ is any net such that $(x_\alpha)$ converges to $\overline{L}_r'(\lambda^+)$ and $\liminf_\alpha y_\alpha>\overline{L}_r'(\lambda^+)$.